(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.
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if 80% of people can read and you have a sample size of 11 people. how many people would you need to interview to be 87% sure that 7 of the 11 people could read
There are 11 people in your sample size and 80% of them can read.
To be 87% sure that 7 of the 11 people could read, you would need to interview eight people.
To calculate this, you need to use a binomial probability formula.
This formula will allow you to calculate the number of successes (people who can read) that must be achieved in order to have an 87% certainty that the sample size of 11 people is representative of the population.
The formula you need to use is P(x) = n! / (x! (n - x)!).
You'll want to plug in x = 7, n = 8. This will give you the probability of 8 successes in 8 trials, which is equal to 0.87. Therefore, you will need to interview 8 people in order to be 87% sure that 7 of the 11 people could read.
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A large bag contains the following marbles:
40 blue marbles
8 green marbles
16 yellow marbles
21 white marbles
Tyra selects a marble from the bag at random. Based on the information, which statement is true?
A.
The correct option for the large bag contains marbles:
A: The marble she selects is twice as likely to be yellow as it is to be green.B: She chooses a marble that has a higher probability of being blue than any other color put together.Explain about the multiple of the number?Multiples are the outcomes of multiplying a numeral by a specific number in mathematics. Examples of multiples of 5 include 10, 15, 20, 25, 30, and cetera.Multiply the integer even by whole number to discover its multiples. For instance, 15 is the third multiple of 5 since 5 3 = 15. The first five multiples of 4 include, for instance, 4, 8, 12, 16, and 20. The first multiple of 4 is 4 since 1 4 = 4.The given question:
marbles in the bag:
40 blue marbles
8 green marbles
16 yellow marbles
21 white marbles
One marble is selected by Tyra.
Then, the correct option are-
A: There is a two to one chance that the marble she chooses will be yellow rather than green.
16 yellow marbles = 2* 8 green marbles
B: Compared to all the other colors combined, the stone she chooses is more likely to be blue.
40 blue marbles have the highest number.
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The complete question is-
A large bag contains the following marbles: •
40 blue marbles
8 green marbles
16 yellow marbles
21 white marbles
Tyra selects a marble from the bag at random. Based on the information, which statement is true?
A The marble she selects is twice as likely to be yellow as it is to be green.
B.The marble she selects is more likely to be blue than all the other colors combined.
C. The marble she selects is equally likely to be yellow or white
D. The marble she selects is three times as likely to be white as it is to be green.
76. cash to find work? (4.2) will cash bonuses speed the return to work of une,ployed people? the illinois department of employment security designed an expierment to find out. the subjects were 10,065
The experiment by the Illinois Department of Employment Security found that cash bonuses can be an effective way to incentivize unemployed people to return to work.
The experiment conducted by the Illinois Department of Employment Security provides evidence on whether cash bonuses can speed up unemployed people's return to work. The results of the experiment suggest that offering cash bonuses can be an effective way to incentivize people to find work.
Specifically, the experiment found that the group of individuals who were offered a $500 cash bonus for finding a job within 11 weeks and holding it for at least 4 months were more likely to find employment than the control group. Additionally, the group that could tell potential employers that the state would pay the employer $500 for hiring them also had a higher likelihood of finding work compared to the control group.
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The given question is incomplete, the complete question is:
The Illinois Department of Employment Security designed an experiment to find out. The subjects were 10,065 people aged 20 to 54 who were filing claims for unemployment insurance. Some were offered $500 if they found a job within 11 weeks and held for at least 4 months. Others could tell potential employers that the state would pay the employer$500 for hiring them. A control group got neither kind of bonus. Will cash bonuses speed unemployed people's return to work?
If you have 4 purple marbles and 8 yellow marbles in a bag, what is
the probability of getting a purple marble ?
Answer: 1/3
Step-by-step explanation:
4 + 8 = 12 so there's 12 marble in the bag and only 4 purple ones. so the probability is 4/12 which we can simplify to 1/3
the sampling distribution becomes approximately normal, for all statistics, if there is a large enough sample size. a. false b. true
The given statement "the sampling distribution becomes approximately normal, for all statistics, if there is a large enough sample size" is True as it is proved by the central limit theorem.
The central limit theorem (CLT) states that for large enough sample sizes, the distribution of the sample means approximates a normal distribution regardless of the shape of the initial population distribution.
It is a statistical theory that clarifies how the mean of a random sample of independent observations drawn from a non-normal population converges in distribution to a normal population.
According to this theory, the sample size should be at least 30 for the sample mean to have an approximately normal distribution. The following are the fundamental assumptions of the central limit theorem:
A large enough sample size is requiredPopulation must be random and independentThe variance of the population must be finiteThe sampling must be done with replacement.Therefore, we can say that the sampling distribution becomes approximately normal, for all statistics, if there is a large enough sample size. The statement is true.
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baby mary recognizes the table as in the same shape, even though the table appears in different shapes depending on the angle from which it is observed. this is an example of
Even though the table appears to have varied shapes depending on the angle from which it is seen, Baby Mary recognizes the table as having the same shape. Shape constancy can be seen in this situation.
What is Shape Constancy?Shape constancy is the ability of an object to retain its shape, size, and orientation even when viewed from different perspectives or angles. In essence, when the shape of an object is known, this allows an individual to recognize the object as a whole.
It's an important part of perception, and it's what allows us to recognize objects even when they're partially obscured or viewed from a different angle. A child can recognize objects as having the same shape, regardless of the angle from which they are viewed. As the object is observed from a variety of angles, its proportions and the shapes of its edges and sides change. Even so, the child perceives the object as having a consistent shape, size, and orientation.
This can be attributed to the ability of a child's brain to maintain a mental image of the object's actual shape, even when presented with an altered image.
As a result, a child may perceive an object as having a consistent shape, even if it appears to be different from one angle to another.
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Find the missing angle measure of the right triangle. Show all work. Round to the nearest tenth.
The missing angle measures 139 degrees.
what is triangle?
A triangle is a polygon that has three sides, three angles, and three vertices. It is a two-dimensional figure that is formed by connecting three non-collinear points. The three sides of a triangle can be of different lengths or the same length, and the angles can also vary in size. The sum of the three angles of a triangle always adds up to 180 degrees. Triangles can be classified based on their side lengths and angle measurements.
In a right triangle, the sum of the two acute angles is always equal to 90 degrees. Therefore, we can find the missing angle by subtracting the two given angles from 90 degrees.
Let's call the missing angle "x". Then we have:
x + 21 + 20 = 180 (the sum of angles in any triangle is 180 degrees)
x = 180 - 21 - 20
x = 139 degrees
Therefore, the missing angle measures 139 degrees.
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which of the following is the correct definition of multicollinearity? group of answer choices when a scatterplot between a single x and a single y shows a straight line, we have a collinearity of 1 when several variables are related, it is fair to assume that each of the related variables account for an identical amount of variation when several variables are related to each other, it is very likely that more than one of these variables does not account for much additional variation we can perform statistical tests with more than two x variables at a time because this will always result in multicollinearity
The statement that shows the correct definition of multicollinearity is that when several variables are related to each other, it is very likely that more than one of these variables does not account for much additional variation. That is option C.
What is multicollinearity?Multicollinearity is defined as the phenomenon that shows the correlative relationship that exists between independent variables in statistics.
The occurrence of multicollinearity found between independent variables is that it will result in less reliable statistical inference and a problem because independent variables should be independent.
Also, more than one of these variables does not account for much additional variation because, changes in one variable are associated with shifts in another variable.
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Classify the expression: 5x3 + 2x − 3
Answer:
12+2x
Step-by-step explanation:
5×3= 15
15+ 2x - 3
collect like terms
15-3+2x
12+2x
therefore 5×3+ 2x - 3 = 12 +2x
Write the equation of the parabola which has its vertex at (0, 5) and passes through the point (1, 0)
y = -5x² + 5 is the equation of the parabola which has its vertex at (0, 5) and passes through the point (1, 0)
We know that the vertex of the parabola is (0, 5), which means that the equation for the parabola has the form:
y = a(x - 0)² + 5
where 'a' is a constant that determines the shape of the parabola. Since the parabola passes through the point (1, 0), we can substitute these values into the equation and solve for 'a':
0 = a(1 - 0)² + 5
0 = a + 5
a = -5
Therefore, the equation of the parabola is: y = -5x² + 5
This equation represents a parabola that opens downwards (since the coefficient of x² is negative), has a vertex at (0, 5), and passes through the point (1, 0).
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How much plastic wrap will be needed to completly cover an ice cream cone that has the slant height of 4,25 inches and a diameter of 2 inches
13.4 square inches of plastic wrap will be needed to completely cover an ice cream cone that has a slant height of 4.25 inches and a diameter of 2 inches.
To completely cover an ice cream cone that has the slant height of 4.25 inches and a diameter of 2 inches, a plastic wrap of about 13.4 square inches will be needed.
How to calculate the amount of plastic wrap required to cover the ice cream cone?
To calculate the amount of plastic wrap required to cover the ice cream cone, we need to find the lateral area of the cone. This can be calculated using the following formula:
Lateral area of cone = πrℓ
where r is the radius of the base of the cone, ℓ is the slant height, and π is the constant pi, which is approximately equal to 3.14.
In this problem, the diameter of the cone is given as 2 inches, which means the radius of the cone will be half of that, i.e., 1 inch. And the slant height is given as 4.25 inches. Therefore, we can calculate the lateral area of the cone as follows: Lateral area of cone = πrℓ = π × 1 × 4.25= 13.4 square inches.
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the distribution of test scores shows that most students did not do well on the test; some did fine and only a few did exceedingly well. the distribution is likely to be .
The distribution of test scores shows that most students did not do well on the test; some did fine and only a few did exceedingly well. The distribution is likely to be skewed.
What is a skewed distribution?
A skewed distribution is one in which the data is not uniformly distributed but rather concentrated on one side. In simple words, a skewed distribution is one in which one tail has a greater number of observations than the other tail, with the majority of data falling in the tail with the greater number of observations.
In a symmetric distribution, the data is uniformly distributed on both sides of the median, with half of the data lying to the right and half to the left.
Thus, the distribution of test scores shows that most students did not do well on the test; some did fine and only a few did exceedingly well. The distribution is likely to be skewed.
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graph y=5x and y=4x–1
The line for y=5x passes through the origin (0,0) and has a steep slope of 5, meaning it rises 5 units for every 1 unit to the right. The line for y=4x-1 intersects the y-axis at -1 and has a slope of 4, meaning it rises 4 units for every 1 unit to the right.
What is the graph?To graph the equations [tex]y=5x[/tex] and [tex]y=4x–1[/tex] , we can plot their corresponding points on a coordinate plane and connect them to form a line.
For [tex]y=5x[/tex]
|
6 | .
| .
5 | .
| .
4 | *
| .
3 | .
| .
2 | .
| .
1 | .
| .
0 |________________________________________
0 1 2 3 4 5 6 7 8 9
Choose any x value, let's say x=0, and find the corresponding y value by substituting it into the equation: y=5(0) = 0. This gives us the point (0,0).
Choose another x value, let's say x=1, and find the corresponding y value: y=5(1) = 5. This gives us the point (1,5).
Therefore, A steep slope of 5 means that the line for y=5x climbs 5 units for every 1 unit to the right and passes through the origin (0,0). The line for [tex]y=4x-1[/tex] has a slope of 4, which means it rises 4 units for every 1 unit to the right, and it meets the y-axis at -1.
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Find the value of x.
In the figure of circle provided. the value of x is
161 degreesHow to find the value of xIn a circle, equal chords subtends equal arc length.
In the problem it was given that:
chord SU is equal to chord ST hence we have that
x + x + 38 = 360 (angle in a circle)
collecting like terms
2x + 38 = 360
2x = 360 - 38
2x = 322
Isolating x by dividing both sides by 2
2x / 2 = 322 / 2
x = 161
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What is (p-1) when p(x)=-x^5-2x^3+4x-3? use the remainder theorm
When the polynomial p(x)=-x^5-2x^3+4x-3, then the value of p(-1) is 3
The Remainder Theorem is a fundamental concept in algebra that relates the value of a polynomial at a certain point to the remainder of the polynomial's division by a linear factor.
The remainder theorem states that when a polynomial p(x) is divided by (x - a), the remainder is equal to p(a).
In this case, we are asked to find p(-1) when p(x) = -x^5 - 2x^3 + 4x - 3.
To use the remainder theorem, we need to divide p(x) by (x - (-1)), which is the same as (x + 1).
We can perform polynomial long division to find the quotient and remainder
The remainder is -x + 2, which means that p(-1) = -(-1) + 2 = 3.
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Given triangle XYZ with ZX = 30°, ZY= 20° and ZZ = 2aº, what is the value of a?
A. 130
B. 65
C. 40
D. 20
According to the question, we are given a triangle XYZ
where,
∠X = 30° ∠Y = 20° ∠Z = 2a°Angle sum property states that sum of all the angles of triangle is 180°. So by adding all the three angles we will get the required value of a.
∠X + ∠Y + ∠Z = 180°
30° + 20° + 2a° = 180°
50° + 2a° = 180°
2a° = 180° - 50°
2a° = 130°
a° = 130/2
a° = 65°
Therefore, required value of a is 65°
Answer:
B
Step-by-step explanation:
remember : the sum of all angles in a triangle is always 180°.
are you sure the third angle is called ZZ ?
I am rather sure it would be called XY !
so,
180 = ZX + ZY + XY = 30 + 20 + 2a
130 = 2a
a = 130/2 = 65°
parasubstitution on carbocation stability substituent effect. describe electron donating vs electron pulling effect on at the para substitution
Parasubstitution is a type of electrophilic aromatic substitution in which the attacking electrophile is substituted into the para position of the aromatic compound. The stability of the carbocation formed during this process is influenced by the substituents present on the aromatic ring.intermediate.
Reason of effecting on parasubstitution:
This is due to the fact that these substituents provide electron density to the aromatic ring, which stabilizes the carbocation by delocalizing the positive charge throughout the ring.Electron withdrawing substituents (such as -NO2, -CN, -COR, -COOH, -SO3H, etc.) have a negative impact on the stability of the carbocation intermediate. T
This is due to the fact that these substituents withdraw electron density from the aromatic ring, making the carbocation intermediate less stable.The substituent effect on carbocation stability in parasubstitution can be represented by the Hammett equation, which relates the substituent effect to the electronic properties of the substituent.
The equation is given by
:log (kX/kH) = ρσ
where kX and kH are the rate constants for the reaction with and without the substituent, respectively
, ρ is the sensitivity constant, and
σ is the substituent constant.
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I need help pls pls pls
Based on the given circle with center Y, each of the measure include the following:
mBC = 71 degrees.mAB = 142 degrees.AD = 30 units.BD = 30 units.YD = 16 units.DC = 18 units.What is a circle?In Mathematics and Geometry, a circle simply refers to a closed, two-dimensional (2D) curved geometric shape with no edges or corners.
Based on the given circle with center Y and line AD is perpendicular to line DB, we can reasonably infer and logically deduce that arc AC is equal to arc BC;
mAC = mBC = 71 degrees.
mAB = mAC + mBC
mAB = 71 + 71
mAB = 142 degrees.
Since line DC is the perpendicular bisector of line AB, the measure of line AB and line BD can calculated as follows;
AD = BD = AB/2
AD = BD = 60/2
AD = BD = 30 units.
Note: YB is the radius of this circle and the hypotenuse of right-angle triangle YDB, which is equal to 34 units.
In order to determine the length of YD, we would have to apply Pythagorean's theorem as follows;
x² + y² = z²
BD² + YD² = YB²
30² + YD² = 34²
YD² = 1156 - 900
YD = √256
YD = 16 units.
DC = YB - YD
DC = 34 - 16
DC = 18 units.
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the difference between a two digits number and its two digits reverse is 72. how many numbers of this type exist?
The two numbers that fit the description are 27 and 99.
There are two numbers that fit this description. To find them, we can solve the equation:
Difference = Number - Number’s Reverse
Difference = 72
Number - Reverse = 72
Let’s call the two-digit number “x” and its reverse “y”. So, we can rewrite the equation as:
x - y = 72
Now, we can solve the equation using algebraic methods. First, add “y” to both sides of the equation:
x - y + y = 72 + y
x = 72 + y
Next, let’s solve for “y” by subtracting “72” from both sides of the equation:
x - 72 = 72 + y - 72
x - 72 = y
Therefore, we can rewrite the equation as:
x - 72 = y
Now, we can solve for “x” and “y” by plugging in possible values for “x” and “y” and seeing which ones work. Since we’re dealing with two-digit numbers, let’s start with numbers between 10 and 99.
First, let’s try x = 10 and y = 82.
10 - 82 = -72
This doesn’t work, since the difference cannot be negative.
Next, let’s try x = 15 and y = 87.
15 - 87 = -72
Again, this doesn’t work.
Finally, let’s try x = 27 and y = 99.
27 - 99 = -72
This works! So, the two numbers that fit the description are 27 and 99.
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A grocer can buy oranges for $1.65 per pound.
1. Write an equation relating c, the cost; and p, the pounds of oranges.
2. An orange order cost $260.10. How many pounds did the grocer order? (Round to the nearest whole number)
Answer:
157 pounds I'm pretty sure the equation would be 1.65c=p
Step-by-step explanation:
p is 1 because no given varible since p=1 pound 1.65 would equal p they 157 pounds bc 260.10/1.65 gives you 157 rounded
can someone help me on this
Figure1 shows pair of corresponding angles
Figure 2 and 6 shows pair of alternate exterior angles
Figure 5 shows pair of alternate interior angle
Figure3 shows pair of consecutive exterior angles
Figure 4 shows pair of consecutive interior angles
Define Alternate Exterior AnglesA set of angles on the outside of each of the two lines but on the opposing sides of the transversal are known as alternate exterior angles.
Define Alternate Interior AnglesWhen a transversal intersects two parallel lines, the angles created within are equal to the alternative pairings of those pairs.
Define Corresponding Anglesthe angles created when two parallel lines cross any other line, whether they are corresponding or matching corners with the transversal.
In the given figures
Figure1 is showing pair of the corresponding angles
Figure 2 and 6 showing pair of the alternate exterior angles
Figure 5 is showing pair of the alternate interior angle
Figure3 is showing pair of the consecutive exterior angles
Figure 4 is showing pair of the consecutive interior angles.
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Any number that can be written as a decimal, write as a decimal to the tenths place.
Given A = (-3,2) and B = (7,-10), find the point that partitions segment AB in a 1:4 ratio.
The point that partitions segment AB in a 1:4 ratio is (
).
The point that partitions segment AB in a 1:4 ratio is [tex]P = \left(-1, -\frac{2}{5}\right)$[/tex].
How to find the ratio?To find the point that partitions segment AB in a 1:4 ratio, we can use the section formula.
Let P = (x, y) be the point that partitions segment AB in a 1:4 ratio, where AP:PB = 1:4. Then, we have:
[tex]$\frac{AP}{AB} = \frac{1}{1+4} = \frac{1}{5}$$[/tex]
and
[tex]$\frac{PB}{AB} = \frac{4}{1+4} = \frac{4}{5}$$[/tex]
Using the distance formula, we can find the lengths of AP, PB, and AB:
[tex]AP &= \sqrt{(x+3)^2 + (y-2)^2} \\PB &= \sqrt{(x-7)^2 + (y+10)^2} \\\ AB &= \sqrt{(7+3)^2 + (-10-2)^2} = \sqrt{244}[/tex]
Substituting these into the section formula, we have:
[tex]$\begin{aligned}x &= \frac{4\cdot(-3) + 1\cdot(7)}{1+4} = -1 \ y &= \frac{4\cdot2 + 1\cdot(-10)}{1+4} = -\frac{2}{5}\end{aligned}$$[/tex]
Therefore, the point that partitions segment AB in a 1:4 ratio is [tex]P = \left(-1, -\frac{2}{5}\right)$[/tex].
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17 identical tasks are assigned to 7 different people. each task is assigned to exactly one person and there are no restrictions on the number of tasks that can be given to any one person. how many ways are there to assign the tasks?
There are 100,947 ways to assign the tasks to the 7 people.
This problem can be solved using combinations. We need to choose 17 tasks from a total of 17, which can be done in one way, and then assign each of these tasks to one of the 7 people. We can do this by considering all possible combinations of tasks that each person could be assigned.
Let's use the stars and bars method to count the number of ways to distribute the tasks. We can represent the tasks as stars and the people as bars, with each bar representing a separate person. For example, if person 1 is assigned 4 tasks, person 2 is assigned 2 tasks, and person 3 is assigned 1 task,
The number of ways to distribute the tasks is then the number of ways to arrange the 17 stars and 6 bars, which is
(17 + 6) choose 6 = 23 choose 6 = 100947
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The price of a television is dropped by 20% during a sale.After the sale the price is increased by 20%. what is the price of the television after the sale if the original price was 1200
The price of the television after the drop in its sales price would be = 1200
How to calculate the price of the television after deduction of sales discount?The percentage of money that was removed from the cost price for the television = 20%
The original price of the television = 1200
The percentage equivalent of 1200 that is 20 = 20/100×1200/1
= 24000/100
= 240
After the sales 240(20%) is added back to the sale price which show that it remains the same after sales.
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How do you fine the blank for infinitely many solutions
To determine if a system of linear equations has infinitely many solutions, you need to solve the system and check the resulting equations.
Here are the general steps:
Write the system of linear equations in standard form: ax + by = c, where a, b, and c are constants.Use any method (substitution, elimination, or matrix) to solve the system of equations and obtain the solution set.If you get an equation that is always true (such as 0 = 0), then the system has infinitely many solutions.If you get an equation that is always false (such as 0 = 1), then the system has no solution.If you obtain a valid solution for each variable, then the system has a unique solution.
If you have two or more variables and only obtain a solution for some of them, then the system has infinitely many solutions.
It's also important to note that if you have a system with fewer equations than variables, it may also have infinitely many solutions.
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(-3m+3d):3 пожалуйста срочноооооо!!!!!!!!!
Answer:
-m + d
Step-by-step explanation:
(-3m + 3d) : 3 =
(-3m + 3d) x 1/3 =
-m + d
-7 + ◾ = 0 x 11
- 11+=-11
6+(-4+)=(6+(-4)+(-7)
The domain of the function using the attached diagram is given by option A. { x / x = -5, -3, 1, 2, 6 }.
As per the attach diagram,
The mapping in the diagram represents the relation between input values and the output values.
Let us consider all input values are represented by variable 'x'.
And all output values are represented by variable 'y'.
All input values represents the domain of the function.
-5 relates to 4
-3 relates to -9
1 relates to 2
2 relates to -6
6 relates to 0
Here,
Domain of the function is ,
{ x / x = -5, -3, 1, 2, 6 }
Range of the function is ,
{ y / y = 4, -9, 2, -6,0 }
Therefore, the domain of the function is equal to option A. { x / x = -5, -3, 1, 2, 6 }.
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The given question is incomplete, I answer the question in general according to my knowledge:
A mapping diagram shows a relation, using arrows, between inputs and outputs for the following ordered pairs: (negative 6, 0), (2, 1), (negative 7, negative 4), (11, 2), (3, 2).
What is the domain of the relation?
Diagram is attached.
PLEASE ANSWER ASAP!!
In triangle ΔABC angle m∠A = 38.68°
What is an angle?An angle is the space between of the point of intersection of two rays meeting at a common vertex.
Since we have triangle ΔABC and m∠B = 90 and the ratio BC/AC = 0.625. We desire to find angle A, we proceed as follows.
In ΔABC, since m∠B = 90, this implies that ΔABC is a right-angled triangle.
Also, the ratio BC/AC = cosC
Since BC/AC = 0.625
⇒ cosC = 0.625
Taking inverse cosine of both sides, we have
C = cos⁻¹(0.625)
= 51.318°
≅ 51.32°
Also in ΔABC
m∠A + m∠B + m∠C = 180° (sum of angles in a triangle)
So, m∠A = 180° - m∠B - m∠C
= 180° - 90° - 51.32°
= 90° - 51.32°
= 38.68°
So, angle m∠A = 38.68°
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water is being drained from a container which has the shape of a right circular cone. the cone has a radius of 3 inches at the top and a height of 6 inches. when the water is 3 inches deep, the surface level is falling at a rate of 2 inches per second. find the rate at which the water is being drained from the container.
The rate at which the water is being drained from the container is -12.56 cubic inches per second.
Given that the water is being drained from a container which has the shape of a right circular cone. The cone has a radius of 3 inches at the top and a height of 6 inches. When the water is 3 inches deep, the surface level is falling at a rate of 2 inches per second. We need to find the rate at which the water is being drained from the container.
Step 1: Calculate the volume of the cone. Consider the cone with the height h = 6 inches and radius of the base r = 3 inches. Let H be the height of the water and V be the volume of the water in the cone. Let V0 be the volume of the cone.
Cone Volume formula: $V=\frac{πr^{2}h}{3}$V0 = πr²h/3 = (3.14) (3²) (6)/3 = 56.52 cubic inches
Step 2: Find the rate at which the water level is falling. Let the rate at which the water level is falling be dh/dt. When the water is 3 inches deep, the surface level is falling at a rate of 2 inches per second. Thus, dh/dt = -2.
Step 3: Find the rate at which the water is being drained from the containerLet dv/dt be the rate at which the water is being drained from the container. To find this rate we need to differentiate the volume of water with respect to time V = (1/3)πr²H.H²dv/dt = (2/3)πr² dh/dtdv/dt = (2/3)π(3²)(-2) = -12.56 cubic inches per second.
Hence, the rate at which the water is being drained from the container is -12.56 cubic inches per second.
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lisa and daisy work at a hair salon. the salon charges $18 fir a hair styling session with lisa and $12 for a session with daisy. the income on a certain day is projected to be $216. this situation can be represented by the equation 18x+12y=216, where x is the number of lisa's customers and y is the number of daisy's customers would lisa need to serve to attain the projected income if daisy calls in sick that day?
Lisa would need to serve 12 customers to attain the projected income if Daisy calls in sick that day
What is linear function ?
A linear function is a mathematical function that can be represented by a straight line on a graph. It has the form of:
y = mx + b
where m is the slope of the line, which determines how steeply the line rises or falls, and b is the y-intercept, which is the point where the line crosses the y-axis.
According to the question:
If Daisy calls in sick, then Lisa will be the only stylist at the salon, and all customers will go to her. Let's use x to represent the number of Lisa's customers. Then the income generated by Lisa's customers can be represented by the equation:
18x = projected income
We can solve for x by dividing both sides of the equation by 18:
x = projected income / 18
Substituting the given projected income of $216, we get:
x = 216 / 18 = 12
Therefore, Lisa would need to serve 12 customers to attain the projected income if Daisy calls in sick that day.
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