Answer: 6
Step-by-step explanation:
The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42
The factors of 78 are: 1, 2, 3, 6, 13, 26, 39, 78
Then the greatest common factor is 6.
Heres something you need to learn about the greatest common factor (gcf)
What is the Greatest Common Factor?
The largest number, which is the factor of two or more numbers is called the Greatest Common Factor (GCF). It is the largest number (factor) that divide them resulting in a Natural number. Once all the factors of the number are found, there are few factors that are common in both. The largest number that is found in the common factors is called the greatest common factor. The GCF is also known as the Highest Common Factor (HCF)
Let us consider the example given below:
Greatest Common Factor (GCF)
For example – The GCF of 18, 21 is 3. Because the factors of the number 18 and 21 are:
Factors of 18 = 2×9 =2×3×3
Factors of 21 = 3×7
Here, the number 3 is common in both the factors of numbers. Hence, the greatest common factor of 18 and 21 is 3.
Similarly, the GCF of 10, 15 and 25 is 5.
How to Find the Greatest Common Factor?
If we have to find out the GCF of two numbers, we will first list the prime factors of each number. The multiple of common factors of both the numbers results in GCF. If there are no common prime factors, the greatest common factor is 1.
Finding the GCF of a given number set can be easy. However, there are several steps need to be followed to get the correct GCF. In order to find the greatest common factor of two given numbers, you need to find all the factors of both the numbers and then identify the common factors.
Find out the GCF of 18 and 24
Prime factors of 18 – 2×3×3
Prime factors of 24 –2×2×2×3
They have factors 2 and 3 in common so, thus G.C.F of 18 and 24 is 2×3 = 6
Also, try: GCF calculator
GCF and LCM
Greatest Common Factor of two or more numbers is defined as the largest number that is a factor of all the numbers.
Least Common Multiple of two or more numbers is the smallest number (non-zero) that is a multiple of all the numbers.
Factoring Greatest Common Factor
Factor method is used to list out all the prime factors, and you can easily find out the LCM and GCF. Factors are usually the numbers that we multiply together to get another number.
Example- Factors of 12 are 1,2,3,4,6 and 12 because 2×6 =12, 4×3 = 12 or 1×12 = 12. After finding out the factors of two numbers, we need to circle all the numbers that appear in both the list.
Greatest Common Factor Examples
Example 1:
Find the greatest common factor of 18 and 24.
Solution:
First list all the factors of the given numbers.
Factors of 18 = 1, 2, 3, 6, 9 and 18
Factors of 24 = 1, 2, 3, 4, 6, 8, 12 and 24
The largest common factor of 18 and 24 is 6.
Thus G.C.F. is 6.
Example 2:
Find the GCF of 8, 18, 28 and 48.
Solution:
Factors are as follows-
Factors of 8 = 1, 2, 4, 8
Factors of 18 = 1, 2, 3, 6, 9, 18
Factors of 28 = 1, 2, 4, 7, 14, 28
Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
The largest common factor of 8, 18, 28, 48 is 2. Because the factors 1 and 2 are found all the factors of numbers. Among these two numbers, the number 2 is the largest numbers. Hence, the GCF of these numbers is 2.
If you know it, dont read it
There are many cylinders with a radius of 6 meters. Let h represent the height in meters and v represent the volume in cubic meters. Write an equation that represents the volume, v , as a function of the height, h
The equation that represents the volume, v, as a function of the height, h is given as: V(h) = 36πh,
where h is the height in meters and V is the volume in cubic meters.
The volume of a cylinder can be calculated using the formula V=πr²h.
Here, we have a number of cylinders with a radius of 6 meters.
Let h represent the height in meters and v represent the volume in cubic meters.
To write an equation that represents the volume, v,
as a function of the height, h,
we can substitute the value of r (radius) with 6m in the formula of the cylinder’s volume,
V = πr²h.
So we get:
V = π(6m)²h
V = 36πh
This equation tells us that the volume of any cylinder with a radius of 6 meters can be expressed as 36πh,
where h is the height of the cylinder.
For similar question on cylinder’s volume.
https://brainly.com/question/27033747
#SPJ11
assume that the relationship between test scores and the student-teacher ratio can be modeled as a linear function with an intercept of 698.9 and a slope of (-2.28). a decrease in the student-teacher ratio by 2 will:
the relationship between test scores and the student-teacher ratio can be modeled as a linear function: The right choice is D) lessen test scores by 4.56 for each school locale.
In view of the data in the inquiry, the relationship between test scores and the understudy educator proportion can be numerically composed as follows:
x = 698.9 - 2.28y .................... (1)
Where,
x = test scores
y = understudy educator proportion
The slant of (- 2.28) shows the sum by which x will change at whatever point there is an adjustment of y.
Hence, when there is a reduction in the understudy educator proportion by 2 (for example y = 2), we will have:
Change in x = - 2.28 * 2 = - 4.56
The negative sign thusly shows that the test scores will decrease by 4.56 for each school area. Accordingly, the right choice is D) lessen test scores by 4.56 for each school locale.
to know more about linear functions click here:
https://brainly.com/question/2248255
#SPJ4
the complete question is:
Assume that the relationship between test scores and the student-teacher ratio can be modeled as a linear function with an intercept of 698.9 and a slope of (-2.28). A decrease in the student teacher ratio by 2 will: A) reduce test scores by 2.28 on average) result in a test score of 698.9C) reduce test scores by 2.56 on average) reduce test scores by 4.56 for every school district.
How much more interest do vour parents have to pay at the end of the first month because their (i point) rating 15 good rather than excellent?
The more interest do your parents have to pay at the end of the first month because their rating is good rather than excellent is option (D) $19.02.
First, we need to calculate the sales tax on the mobile home purchase:
Sales tax = 4.2% of $89,000 = 0.042 x $89,000 = $3,738
The total cost of the mobile home including sales tax is
Total cost = $89,000 + $3,738 = $92,738
After making a down payment of $3,000, the amount to be financed is:
Amount financed = $92,738 - $3,000 = $89,738
To calculate the interest, we need to know the monthly interest rate. Let's assume the loan term is 30 years, which is 360 months. Then, the monthly interest rate for an excellent credit rating would be:
Monthly interest rate (excellent) = 4.75 / 1200 = 0.00396
And for a good credit score
Monthly interest rate (good) = 5.00 / 1200 = 0.00417
The interest for the first month for an excellent credit rating would be:
Interest (excellent) = $89,738 x 0.00396 = $355.33
And for a good credit rating:
Interest (good) = $89,738 x 0.00417 = $374.43
The difference in interest is:
Difference in interest = $374.43 - $355.33 = $19.02
Therefore, the correct option is (D) $19.02.
Learn more about credit score here
brainly.com/question/2272694
#SPJ4
The given question is incomplete, the complete question is:
Your parents are purchasing a mobile home for $89,000. The sales tax is 4.2%, they make a $3,000 down payment. How much more interest do your parents have to pay at the end of the first month because their rating is good rather than excellent?
Secured Unsecured
Credit APR (%) APR (%)
Excellent 4.75 5.50
Good 5.00 5.90
Average 5.85 6.75
Fair 6.40 7.25
Poor 7.50 8.40
Options:
A) $29.39
B) $21.10
C) $18.71
D) $19.02
traveling for 3 hr into a steady headwind, a plane flies 1650 mi. the pilot determines tha tflying with the same wind for 2 hr, he could make a trip of 1300. find the rate of the plane and the wind speed
The rate of the plane (in still air) is 600 miles per hour and the speed of the wind is 50 miles per hour.
Let's denote the rate of the plane (in still air) by p and the speed of the wind by w.
From the first part of the problem, we know that the plane flies 1650 miles in 3 hours against the headwind. This means that the effective speed of the plane (relative to the ground) was 1650/3 = 550 miles per hour slower than its speed in still air, i.e., p - w = 550.
From the second part of the problem, we know that the plane could make a trip of 1300 miles in 2 hours with the same wind. This means that the effective speed of the plane (relative to the ground) was 1300/2 = 650 miles per hour faster than its speed in still air, i.e., p + w = 650.
Now we have two equations with two unknowns, which we can solve using a system of equations. Adding the two equations, we get:
2p = 1200
Therefore, p = 600 miles per hour. Substituting this value into one of the equations, we get:
600 + w = 650
Therefore, w = 50 miles per hour.
Learn more about Speed here:
https://brainly.com/question/2004627
#SPJ4
a bond is worth 100$ and grows in value by 4 percent each year. f(x) =
To represent the value of the bond after x years, we can use the function:f(x) = 100 * (1 + 0.04)^xwhere x is the number of years the bond has been held.The expression (1 + 0.04) represents the growth factor of the bond per year, since the bond grows in value by 4 percent each year. By raising this factor to the power of x, we obtain the cumulative growth of the bond over x years.Multiplying the initial value of the bond, 100$, by the growth factor raised to the power of x, gives us the value of the bond after x years. This is the purpose of the function f(x).
explain what the p-value means in the given context. a state university wants to increase its retention rate of 4% for graduating students from the previous year. after implementing several new programs during the last two years, the university reevaluates its retention rate and comes up with a p-value of 0.075. using , what can we conclude?
In this context, the p-value represents the probability of observing the data or a more extreme result, assuming that the null hypothesis (the retention rate is still 4%) is true.
In other words, it measures the strength of evidence against the null hypothesis. A small p-value indicates that the observed result is unlikely to have occurred by chance alone, while a large p-value suggests that the null hypothesis cannot be rejected.
In this case, the calculated p-value of 0.075 suggests that there is some evidence to reject the null hypothesis that the retention rate is still 4%.
However, since the p-value is above the conventional threshold of 0.05, we cannot conclude with certainty that the new programs have significantly increased the retention rate. Instead, we can say that there is some indication that the programs may have been effective, but further investigation is needed to determine if this is true.
To learn more about p-value click on,
https://brainly.com/question/29675177
#SPJ4
are the following statements true or false? if false, correct the statement. a. the precision of your answer is determined by the largest place of significance when adding and subtracting.
The statement "the precision of your answer is determined by the largest place of significance when adding and subtracting" is False because The precision of your answer is determined by the number of decimal places when adding and subtracting.
For example, if one number is given to the nearest hundredth and the other is given to the nearest tenth, then the result should be given to the nearest tenth. This is because the tenths place is the largest place of significance in this case.
When adding and subtracting, it is important to take into account the precision of each number before adding or subtracting.
If one number is given to the nearest tenth and the other is given to the nearest hundredth, then the result should be given to the nearest hundredth, as the hundredths place is the largest place of significance.
The same applies for any other number of decimal places.
To know more about decimal places click on below link:
https://brainly.com/question/50455#
#SPJ11
A future interest usually exists in conjunction with which of the following real property interests at the time that it is the interest granted?
A future interest in real property typically exists in conjunction with a present interest in that same property at the time the future interest is granted.
What is a future interest?
Future interests are created when a property owner grants an interest in their property that will take effect at a later time, such as after their death or the expiration of a lease.
These interests are typically created in the form of a trust or a will, and they can be used to ensure that property is distributed according to the owner's wishes, to provide for the future needs of family members, or to protect property from creditors or other claims.
Examples of future interests include remainders, reversions, and executory interests.
Complete questin:
A future interest usually exists in conjunction with which of the following real property interests at the time that it is the interest granted?
1) future interest
2) protect property
3) both a and b
4) None of these
To know more about Interest:
https://brainly.com/question/13324776
#SPJ11
Solve for n: 8л = 56
n/8⋅л=56. You’re welcome!
Oliver spots an airplane on radar that is currently approaching in a straight line, and
that will fly directly overhead. The plane maintains a constant altitude of 6900 feet.
Oliver initially measures an angle of elevation of 16° to the plane at point A. At some
later time, he measures an angle of elevation of 27° to the plane at point B. Find the
distance the plane traveled from point A to point B. Round your answer to the
nearest tenth of a foot if necessary.
Thus, the distance travelled by the airplane from point A to point B is found as: 1159.93 ft.
Explain about the angle of elevation?The angle from of the horizontal upward to an item is referred to as the angle of elevation. The line of sight of an observer will rise above the horizontal.The angle from horizontal downward to an item is referred to as the "angle of depression." The line of sight of an observer would fall short of the horizontal.Let the distance travelled between A and B be 'x'.
Height h = 6900 feet.
The figure is attached.
Using the tan function:
tan Ф = opposite side/ hypotenuse
In right triangle BCE.
tan 27 = 6900 / y
y = 6900 / tan 27
y = 15281.80
In right triangle ACD
tan 16 = 6900 /(x + y)
x + y = 6900/tan 16
x = 26873.72 - 15281.80
x = 1159.93
Thus, the distance travelled by the airplane from point A to point B is found as: 1159.93 ft.
Know more about the angle of elevation
https://brainly.com/question/15580615
#SPJ1
during the summer of 2018, coldstream country club in cincinnati, ohio collected data on 443 rounds of golf played from its white tees. the data for each golfer's score on the twelfth hole are contained in the datafile coldstream12. (round your answers to four decimal places.) (a) construct an empirical discrete probability distribution for the player scores on the twelfth hole.
Given information:
During the summer of 2018, Coldstream Country Club in Cincinnati, Ohio collected data on 443 rounds of golf played from its white tees. The data for each golfer's score on the twelfth hole are contained in the datafile coldstream12. (Round your answers to four decimal places.)
Construct an empirical discrete probability distribution for the player scores on the twelfth hole.
A probability distribution is a tabulation of probabilities for every outcome in a sample space. It is a summary of the probabilities for all possible values of a random variable. In statistics, there are two types of probability distributions: empirical and theoretical.
An empirical distribution is a frequency distribution of observed data. A theoretical distribution is a distribution that is derived from theory.
To construct the empirical probability distribution, we will use the following steps:
Step 1: Calculate the range of the data set, which is the difference between the maximum and minimum values. Range = Maximum value – Minimum value = 8 – 3 = 5.
Step 2: Divide the range by the number of intervals (or classes) desired. In this case, we will use 6 intervals. Interval size = Range / Number of intervals = 5 / 6 = 0.833333333.
Step 3: Determine the lower limit of each interval by subtracting the interval size from the minimum value.
Lower limit of first interval = Minimum value – Interval size = 3 – 0.833333333 = 2.166666667.
Lower limit of second interval = Lower limit of first interval + Interval size = 2.166666667 + 0.833333333 = 2.999999999 ≈ 3.
Lower limit of third interval = Lower limit of second interval + Interval size = 3 + 0.833333333 = 3.833333333.
Lower limit of fourth interval = Lower limit of third interval + Interval size = 3.833333333 + 0.833333333 = 4.666666666.
Lower limit of fifth interval = Lower limit of fourth interval + Interval size = 4.666666666 + 0.833333333 = 5.499999999 ≈ 5.
Lower limit of sixth interval = Lower limit of fifth interval + Interval size = 5 + 0.833333333 = 5.833333333.
Step 4: Count the number of data points that fall in each interval. The table below shows the frequency distribution for the data set.
Interval Lower Limit Upper Limit Frequency Relative Frequency
1 2.166666667 2.999999999 33 0.0745
2 3 3.833333333 93 0.2096
3 3.833333333 4.666666666 74 0.1669
4 4.666666666 5.499999999 56 0.1264
5 5.5 6.333333333 8 0.0180
6 5.833333333 6.666666666 4 0.0090
Total - - 268 1.0000
From the above table, the empirical discrete probability distribution is constructed.
To learn more about empirical, refer below:
https://brainly.com/question/30439049
#SPJ11
Solve for x. Round to the nearest tenth, if necessary.
D
59⁰
7.7
E
Xx
F
Click her
The value of x in the right triangle when calculated is approximately 13.8 units
Calculating the value of x in the triangleGiven the right-angled triangle
The side length x can be calculated using the following sine ratio
So, we have
sin(39) = x/22
To find x, we can use the fact that sin(39 degrees) = x/22 and solve for x.
First, we can use a calculator to find the value of sin(39 degrees), which is approximately 0.6293.
Then, we can set up the equation:
0.6293 = x/22
To solve for x, we can multiply both sides by 22:
0.6293 * 22 = x
13.8446 = x
Rewrite as
x = 13.8446
Approximate the value of x
x = 13.8
Therefore, x is approximately 13.8 in the triangle
Read about right triangle at
https://brainly.com/question/2437195
#SPJ1
Bo and Erica are yoga instructors. Between the two of them, they teach 37 yoga classes each week. If Erica teaches 14 fewer than
twice as many as Bo, how many classes does each instructor teach per week?
OA. 19 Bo; 18 Erica
OB. 14 Bo; 23 Erica
OC. 21 Bo; 16 Erica
OD. 17 Bo; 20 Erica
Answer:
The correct answer is OD.
Step-by-step explanation:
To find how many classes each instructor teaches per week, you need to set up and solve a system of equations based on the given information. In other words,
Let x be the number of classes Bo teaches per week and y be the number of classes Erica teaches per week.
The first equation is based on the total number of classes they teach
x + y = 37
The second equation is based on the relationship between their numbers of classes
y = 2x - 14
To solve this system, you can use substitution or elimination methods. For example, using substitution, you can plug in y = 2x - 14 into the first equation and get
x + (2x - 14) = 37
Simplify and solve for x
3x - 14 = 37
So, Bo teaches 17 classes per week and Erica teaches 20 classes per week.
Answer:
A. 19 Bo; 18 Erica
Step-by-step explanation:
im good like that hehe
The prisms are similar. What is the surface area of Prism B? Prism A is 10 m. Prism B is 6 m. Surface area = 880m2
Answer:
79.92m
Step-by-step explanation:
here you go hope this helps
Find the exact value of X.
In the given right angled triangle, using Pythagorean theorem, the value of x is 12.73 units.
What is Pythagorean theorem?The Pythagorean theorem, also known as Pythagoras' theorem, is a basic relationship between a right triangle's three sides in Euclidean geometry. According to this rule, the length of the squares on the other two sides add up to the length of the square whose side is the hypotenuse, or the side across from the right angle. This theory can be expressed as the Pythagorean equation, which is an equation relating the lengths of the sides a, b, and the hypotenuse c:
a² + b² =c²
What is Hypotenuse?The longest side of a right-angled triangle, or the side across from the right angle, is called the hypotenuse.
In the given right angled triangle, using Pythagorean theorem,
x² = (9[tex]\sqrt{2}[/tex] )²+ (9[tex]\sqrt{2}[/tex] )²
x² = 81×2
x = [tex]\sqrt{162}[/tex]
x = 12.73 units
Therefore, the value of x will be 12.73 units.
To know more about Pythagorean Theorem, visit
https://brainly.com/question/343682
#SPJ1
in california, people use 63 btu of energy per home. this is 30% less than the national average. how many btu does an average american household use?
The national average for energy usage per home in BTU is approximately 90 BTU.
BTU stands for British Thermal Unit, which is the amount of energy required to raise the temperature of one pound of water by one degree Fahrenheit. It is used to quantify the amount of energy consumed by heating or cooling equipment.In California, people use 63 BTU of energy per home, which is 30% less than the national average. Therefore, to calculate the average BTU consumption for American households, we will use the following formula: National average BTU = California BTU / (1 - 30%)
Let's solve for the national average BTU: National average BTU = 63 BTU / (1 - 0.30)
National average BTU = 63 BTU / 0.70National average BTU = 90 BTU
Therefore, the average American household uses 90 BTU of energy per home.
Learn more about percentages and linear equations:https://brainly.com/question/28654618
#SPJ11
Isla will deposit $800 in an account that earns 5% simple interest every year. Her sister Aven will deposit $750 in an account that earns 7% interest compounded annually. The deposits will be made on the same day, and no additional money will be deposited or withdrawn from the accounts. Which statement is true about the balances of the girls’ accounts at the end of 48 months?
To compare the balances of the two accounts, we need to calculate the future value of each account after 48 months.
For Isla's account with simple interest, the interest earned every year is:
$800 x 0.05 = $40
After 48 months (4 years), the interest earned is:
$40 x 4 = $160
So the balance of Isla's account after 48 months is:
$800 + $160 = $960
For Aven's account with compound interest, we can use the formula:
A = P(1 + r/n)^(nt)
where A is the future value, P is the principal (initial deposit), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time in years.
For Aven's account, P = $750, r = 0.07, n = 1 (compounded annually), and t = 4. Plugging these values into the formula, we get:
A = $750(1 + 0.07/1)^(1 x 4) = $1039.14
Therefore, the statement "Aven's account will have a higher balance than Isla's account at the end of 48 months" is true.
There is a system of two linear equations. The first equation is y=7x+8. The system has infinite solutions. What other equation would complete the system?
If the system has infinite solutions, it means that the two equations are not independent, but rather one equation is a multiple of the other. In other words, the second equation must be a multiple of y = 7x + 8.
One possible equation that completes the system is:
14y = 98x + 112
To check that this equation works, we can substitute y = 7x + 8 into the equation:
14(7x + 8) = 98x + 112
98x + 112 = 98x + 112
As we can see, the equation is true for any value of x, which means that it is satisfied for any value of y that corresponds to y = 7x + 8.
Therefore, the system of equations that has y = 7x + 8 as one of its equations and has infinite solutions is:
y = 7x + 8
14y = 98x + 112
If the dartboard has a diameter of 14 inches and each number occupies 1/20 of the board, how much space does a single number occupy in square inches?
Each number occupies 7.7 inches² of the dartboard.
What is area?In geοmetry, area is the amοunt οf space a flat shape, like a pοlygοn, circle οr ellipse, takes up οn a plane. The area οf a shape is always measured in square units. Tο find the area οf simple shapes like a square οr the area οf a rectangle, yοu οnly need its width, w, and length, l (οr base, b). The area is length times width:
First we need the area of the dartboard
Area = πr²
here r = d/2 = 14/2 = 7
Area of dartboard = πr²
[tex]$ \rm = \frac{22}{7 } \times 7 \times 7[/tex]
= 22 × 7
= 154 inches²
Now, its is said that each number occupies 1/20 of the board,
Then area of each number = 154 × 1/20
= 7.7 inches²
Thus, Each number occupies 7.7 inches² of the dartboard.
learn more about area
https://brainly.com/question/27683633
#SPJ1
Vincent flipped three coins during a probability experiment. The outcomes of the first 40 trials are shown
Based on the information in the table, in how many of the next 120 trials will the outcome be exactly two of the coins showing heads? in the table.
Answer:
I'm sorry, but I cannot see the table you are referring to. However, I can explain the general approach to solving this type of problem.
When flipping a fair coin, there are two possible outcomes: heads or tails. The probability of getting heads on any one flip is 1/2, and the probability of getting tails is also 1/2.
If Vincent has already flipped the coins 40 times and recorded the outcomes, we can use that information to estimate the probability of getting exactly two heads in the next 120 flips.
One way to do this is to find the proportion of the 40 trials that resulted in exactly two heads. Let's say that out of the 40 trials, 10 of them resulted in exactly two heads. Then the proportion of trials that resulted in two heads is:
10/40 = 1/4
This means that the probability of getting exactly two heads in one trial is 1/4.
To find the expected number of trials out of the next 120 that will result in exactly two heads, we can multiply the probability of getting two heads in one trial by the total number of trials:
(1/4) x 120 = 30
Therefore, we would expect that out of the next 120 trials, approximately 30 of them would result in exactly two heads. However, this is just an estimate based on the information given. The actual number of trials that result in exactly two heads could be more or less than 30 due to random variation.
the probability that a dessert sold at a certain cafe contains chocolate is 86%. the probability that a dessert containing chocolate also contains nuts is 30%. find the probability that a dessert chosen at random contains nuts given that it contains chocolate. round to the nearest tenth of a percent.
The probability that a dessert chosen at random contains nuts given that it contains chocolate is 34.9%.
The likelihood of any event A happening when another event B related to A has already happened is known as conditional probability. This implies that the likelihood of event A relies on event B.
The conditional probability symbol is P(A|B). Conditional probability P(A|B) is crucial for any exam that is part of a competition. The concept of conditional probability P(A|B), formula, and examples with solutions are all covered in this article. The Bayes theorem predicts the likelihood that an event connected to any condition would occur. For the situation of conditional probability, this theorem is taken into account.
Let A : a certain cafe contains chocolates
B: a dessert contains Nuts
P(A) = 86% = 0.86
P(AB) = 30% = 0.3
[tex]P(B|A) =\frac{P(AB)}{P(A)}[/tex]
= 0.3/0.86
= 34.9 %
So the answer is 34.9%
Learn more about Probability:
https://brainly.com/question/2364330
#SPJ4
3. Consider the following statement: For each positive real number r, if r2 = 18, then r is irrational (a) If you were setting up a proof by contradiction for this statement, what would you assume? Carefully write down all conditions that you would assume. (b) Complete a proof by contradiction for this statement.
(a) To set up a proof by contradiction for the statement "For each positive real number r, if r^2 = 18, then r is irrational," we will assume the opposite of the given statement. Specifically, we would assume that there exists a positive real number r such that r^2 = 18, and r is rational (not irrational).
(b) To complete the proof by contradiction, follow these steps:
1. Assume that there exists a positive real number r such that r^2 = 18, and r is rational.
2. Since r is rational, we can write it as a fraction a/b, where a and b are integers with no common factors other than 1 (i.e., a and b are coprime) and b ≠ 0.
3. We have r^2 = 18, so (a/b)^2 = 18.
4. Squaring both sides, we get a^2 / b^2 = 18.
5. Rearrange the equation: a^2 = 18b^2.
6. Since 18 is an even number, a^2 is also an even number, which implies that a is an even number (let's say a = 2k, where k is an integer).
7. Substitute a with 2k: (2k)^2 = 18b^2, which simplifies to 4k^2 = 18b^2.
8. Divide both sides by 2: 2k^2 = 9b^2.
9. Now, we see that the left side of the equation is an even number (2k^2), which implies that 9b^2 is also an even number. However, 9 is an odd number, so for 9b^2 to be even, b must be even.
10. Both a and b are even, which contradicts our original assumption that a and b are coprime (having no common factors other than 1).
Therefore, our assumption that there exists a positive real number r such that r^2 = 18 and r is rational leads to a contradiction. Thus, the original statement is true: for each positive real number r, if r^2 = 18, then r is irrational.
To know more about Rational and Irrational numbers:
https://brainly.com/question/17450097
#SPJ11
how do we ensure trigonometric functions compute values appropriately? what is the value of this expression?
We can ensure trigonometric functions compute values appropriately by validating the results against known values. In order to ensure that the argument is in a short interval surrounding a place where an approximation is highly accurate, one performs a range reduction prior to computing things like a cos or sin.
By comparing the results to established values, we can make sure trigonometric functions compute values correctly. As an example, if we wanted to validate the cosine of 45 degrees, we could compare the result to the known value of 0.707. If the computed value is within a small margin of error of the known value, then we can assume the trigonometric function is computing values appropriately.
In order to calculate functions like cos or sin, one must first perform a range reduction to make sure the input is inside a narrow range surrounding a point where an approximation is highly accurate. It is typically close to zero.
To learn more about trigonometric functions link is here
brainly.com/question/6904750
#SPJ4
Solve (x – 3)2 = 49. Select the values of x. –46, -4, 10, 52.
Answer:
x = 10, -4
Step-by-step explanation:
(x – 3)² = 49
x - 3 = √49
x - 3 = ± 7
x = 10, -4
Looking at options, 10 and -4 is the answer.
A rectangular garden plot has a length of 4 meters and a width of 3.5 meters. Jackson wants to put a fence around the garden. He also wants to install landscape fabric over the entire plot to prevent weeds. How much fencing will he need to buy? How much landscape fabric will he need to buy?
Using the formula of area and perimeter of the rectangle, the answers to both subparts are shown:
(A) The fencing required is 15 meters.
(B) The fabric required is 14m².
What is a rectangle?An enclosed 2-D shape called a rectangle has four sides, four corners, and four right angles (90°).
A rectangle has equal and parallel opposite sides.
Four sides make up a rectangle, which has opposing sides that are parallel and of equal length.
It belongs to a category of quadrilaterals where each of the four angles is a straight angle or measures 90 degrees.
An example of a parallelogram with equal angles is a rectangle.
It can alternatively be described as a parallelogram with a right angle or an equiangular quadrilateral, where equiangular denotes that all of its angles are equal.
A square is a rectangle with four equally long sides.
(A) The required fencing:
Perimeter = 2(l+w)
Perimeter = 2(7.5)
Perimeter = 15
The fencing required is 15 meters.
(B) The landscape fabric required:
Area = l*w
Area = 4*3.5
Area = 14m²
The fabric required is 14m².
Therefore, using the formula of area and perimeter of the rectangle, the answers to both subparts are shown:
(A) The fencing required is 15 meters.
(B) The fabric required is 14m².
Know more about rectangles here:
https://brainly.com/question/25292087
#SPJ1
An architect is designing a house. He wants the bedroom to have the dimensions of 10 ft by 6 ft by 7 ft. The architect doubles all three dimensions to create the den. Does that mean the den will have double the volume of the bedroom? First, find the volume of the bedroom. Solve on paper. Then check your work on Does doubling all three dimensions mean the den will have double the volume of the bedroom? Why or why not? Does doubling all three dimensions mean the den will have double the volume of the bedroom? No If he doubles all three dimensions, the den's volume will be more than double the volume of the bedroom.
Answer:
(2 x 10) x (2 x 6) x (2 x 7) = 3360ft³ <-- the den would be that
The volume:
10 x 6 x 7 = 420ft³ Volume = 8
license plates are made using 2 letters followed by 2 digits. how many plates can be made if repetition of letters and digits is allowed?
The number of license plates that can be made if repetition of letters and digits is allowed is:
67,600
How to determine the number of plates when repeating the letters and numbers?Can be found by multiplying the number of possibilities for each position.
License plates are made using 2 letters followed by 2 digits. In this case, repetition is allowed, which means that each of the 2 letters can be any of the 26 letters of the English alphabet, and each of the 2 digits can be any of the 10 digits from 0 to 9. Thus, the total number of possible license plates can be found by multiplying the number of possibilities for each position.
There are 26 choices for each of the two letter positions, and 10 choices for each of the two digit positions, so the total number of possible license plates is:
26 x 26 x 10 x 10 = 67,600
Therefore, there are 67,600 plates that can be made if repetition of letters and digits is allowed.
More informationa about Permutations and Combinations here: https://brainly.com/question/28065038
#SPJ11
ginas journal has a square cover with the side length shown. what is the perimeter of the journal? what is the area of the jouranl?
The given is of the square cover of Gina's journal with a side length of 24 cm. So the perimeter of the journal is 96 cm and the area of the journal would be 576 square cm.
If Gina’s journal has a square cover with a side length of 24 cm, then the perimeter of the journal would be 4 times the side length,
Which is 4 x 24 = 96 cm.
To calculate the area of a square, you need to multiply the length of one of its sides by itself.
The formula for the area of a square is,
Area = Side x Side or Area = Side².
Therefore, the area of the journal would be the side length squared,
Which is 24 x 24 = 576 square cm.
Question: Gina's journal has a square cover with the side length shown as 24 cm. what is the perimeter of the journal? what is the area of the journal?
Learn more about the square cover area and perimeter at: https://brainly.com/question/25092270
#SPJ11
what is the probability of rolling a sum less than or equal to 7 ? express your answer as a fraction or a decimal number rounded to four decimal places.
The probability of rolling a sum of 7 or less on two dice is 11/36. This can be expressed as a decimal number rounded to four decimal places as 0.3056.
We need to look at the possibilities when rolling two dice. Each die has six sides, which means the total number of outcomes when rolling two dice is 36. There are 11 combinations of two dice that produce a sum of 7 or less: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (3,1), (3,2). So, 11/36 can be simplified to 1/4, or 0.3056.
To understand this in a more visual way, you can draw a table and use colored dots to indicate the possible combinations. This will show you that out of 36 total possibilities, 11 of them produce a sum of 7 or less.
In summary, the probability of rolling a sum of 7 or less on two dice is 11/36, which can be expressed as a decimal number rounded to four decimal places as 0.3056.
For more such questions on Probability.
https://brainly.com/question/27279592#
#SPJ11
now standardize this distance by dividing it by the population standard-deviation (or standard error). in other words, how many population-standard-deviations (or standard-error) away is your sample mean from the population mean?
Population-standard-deviations (or standard-error) away is your sample mean from the population mean is therefore (x - μ) / σ.
In order to standardize the distance, we must divide it by the standard deviation of the population (or standard error).How far the sample mean is from the population mean in terms of population-standard-deviations (or standard-error) can be determined by dividing it by the population standard deviation (or standard error). The equation for the same is as follows:
z = (x - μ) / σ
where,
z = standard score,
x = sample mean,
μ = population mean,
σ = population standard deviation.
for such more question on population mean
https://brainly.com/question/28103278
#SPJ11