Find the centre and radius of -6x+x^2=97+10y-y^2
Answer:
Centre = (3, 5)
Radius = [tex]\sqrt{131}[/tex]
Step-by-step explanation:
Given equation of a circle:
[tex]-6x + x^2 = 97 + 10y - y^2[/tex]
To find the centre and radius of the given equation of a circle, rewrite it in standard form.
[tex]\boxed{\begin{minipage}{6.3cm}\underline{Equation of a Circle - Standard Form}\\\\$(x-h)^2+(y-k)^2=r^2$\\\\where:\\ \phantom{ww}$\bullet$ $(h, k)$ is the centre. \\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]
First, rearrange the equation so that the terms in x and y are on the left side and the constant is on the right side:
[tex]x^2 - 6x + y^2 - 10y = 97[/tex]
Complete the square for the x and y terms by adding the square of half the coefficient of the term in x and y to both sides:
[tex]\implies x^2 - 6x +\left(\dfrac{-6}{2}\right)^2+ y^2 - 10y +\left(\dfrac{-10}{2}\right)^2= 97+\left(\dfrac{-6}{2}\right)^2+\left(\dfrac{-10}{2}\right)^2[/tex]
Simplify:
[tex]\implies x^2 - 6x +\left(-3\right)^2+ y^2 - 10y +\left(-5\right)^2= 97+\left(-3\right)^2+\left(-5\right)^2[/tex]
[tex]\implies x^2 - 6x +9+ y^2 - 10y +25= 97+9+25[/tex]
[tex]\implies x^2 - 6x +9+ y^2 - 10y +25= 131[/tex]
Now we have created two perfect square trinomials on the left side of the equation:
[tex]\implies (x^2 - 6x +9)+ (y^2 - 10y +25)= 131[/tex]
Factor the perfect square trinomials:
[tex]\implies (x-3)^2+ (y-5)^2= 131[/tex]
If we compare this equation with the standard form, we see that the centre of the circle is (3, 5) and its radius is the square root of 131.
Therefore:
centre = (3, 5)radius = [tex]\sqrt{131}[/tex]Perform the indicated operation. Reduce to lowest terms if possible. 4 1/5 ÷ 2 1/3
The requried, Reduction to the lowest terms of 4 1/5 ÷ 2 1/3 is 9/5.
To divide mixed numbers, we need to convert them to improper fractions, then multiply the first fraction by the reciprocal of the second fraction.
Converting the mixed numbers to improper fractions:
4 1/5 = 21/5
2 1/3 = 7/3
Multiplying by the reciprocal:
(21/5) ÷ (7/3) = (21/5) * (3/7) = 9/5
Therefore, 4 1/5 ÷ 2 1/3 = 9/5.
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Please help with this question!
Answer:
The distance between the library and museum is 400 yards
Step-by-step explanation:
what is the sum of the first 30 terms of the sequence 2,5,8,11,14,17
The first 30 terms of the sequence are 89.
What is an arithmetic sequence?A progression or sequence of numbers known as an arithmetic sequence maintains a consistent difference between each succeeding term and its predecessor. The common difference in that mathematical progression is the constant difference.
Here, we have
Given: the sequence 2,5,8,11,14,17.
We have to find the first 30 terms of the sequence.
It is known that the value of the first term (a) = 2.
Different from the sequence (d) = 5 - 2 = 3.
To solve this problem, we use the formula [tex]$ \rm S_n=(\frac{n}{2})(2a+d(n-1))[/tex]
[tex]$ \rm S_{30}=(\frac{30}{2})(2(2)+(3)(30-1))[/tex]
[tex]$ \rm S_{30}=(15)(4+(3)(29))[/tex]
S₃₀ = 1365
Hence, the first 30 terms of the sequence are 89.
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The pentagonal prism below has a height of 13.4 units and a volume of 321.6 units ^3 . Find the area of one of its bases.
The area of one of the bases of the pentagonal prism is approximately 172.96 square units.
What is Area ?
Area is a measure of the amount of space inside a two-dimensional shape, such as a square, rectangle, triangle, circle, or any other shape that has a length and a width. It is usually measured in square units, such as square inches, square feet, or square meters.
To find the area of one of the bases of the pentagonal prism, we need to use the formula for the volume of a pentagonal prism, which is:
V = (1÷2)Ph,
where V is the volume, P is the perimeter of the base, h is the height of the prism.
Since we know that the height of the prism is 13.4 units and the volume is 321.6 , we can solve for the perimeter of the base:
V = (1÷2)Ph
321.6 = (1÷2)P(13.4)
P = 48
The perimeter of the base is 48 units.
To find the area of one of the bases, we can use the formula for the area of a regular pentagon, which is:
A = (5÷4) [tex]s^{2}[/tex]* tan(π÷5)
where A is the area of the pentagon and s is the length of a side.
Since the pentagon is regular, all sides have the same length. Let's call this length "x".
The perimeter of the pentagon is 48 units, so we have:
5x = 48
x = 9.6
Now we can use the formula for the area of a regular pentagon to find the area of one of the bases:
A = (5÷4)[tex]x^{2}[/tex] * tan(π÷5)
A = (5÷4)(9.6*9.6) * tan(π÷5)
A ≈ 172.96
Therefore, the area of one of the bases of the pentagonal prism is approximately 172.96 square units.
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$5,900 is invested in an account earning 5.6% interest (APR), compounded daily.
Write a function showing the value of the account after t years, where the annual
growth rate can be found from a constant in the function. Round all coefficients in
the function to four decimal places. Also, determine the percentage of growth per
year (APY), to the nearest hundredth of a percent.
The function for the value of the account after t years is: [tex]V(t) = 5900 * (1 + 0.056/365)^{(365*t)[/tex]and the percentage of growth per year is 5.74%.
What is percentage?A percentage is a way of expressing a quantity as a fraction of 100. It is often used to compare two quantities or to express a part of a whole.
In mathematics, a function is a rule that assigns to each element in one set (called the domain) exactly one element in another set (called the range).
According to given information:The formula for calculating the value of the account after t years is:
[tex]V(t) = 5900 * (1 + 0.056/365)^{(365*t)[/tex]
where 0.056 is the annual interest rate (APR), divided by 100 to convert it to a decimal, and 365 is the number of days in a year.
To find the annual percentage yield (APY), we can use the formula:
[tex]APY = (1 + APR/n)^{n - 1[/tex]
where n is the number of times the interest is compounded per year. In this case, the interest is compounded daily, so n = 365.
[tex]APY = (1 + 0.056/365)^{365 - 1} = 0.0574\ or\ 5.74%[/tex]
Therefore, the function for the value of the account after t years is:
[tex]V(t) = 5900 * (1 + 0.056/365)^{(365*t)[/tex]
And the percentage of growth per year is 5.74%.
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This shape is made up of one half-circle attached to a square with side lengths 17 inches. You can use 3.14 as an approximation for pie
Answer:
109.68
Step-by-step explanation:
The side lengths are 24 not 17 according to the directions.
The sides of the square would be 24 + 24 + 24 = 72
Circumference of the 1/2 circle:
c = [tex]\frac{2\pi r}{2}[/tex] We are dividing by 2 because we have 1/2 of a circle
c= [tex]\frac{(2)(3,14)(12)}{2}[/tex] The diameter is 24 so the radius is 12
c = 37.68
Add 72 + 37.68 = 109.68
Helping in the name of Jesus.
it takes brian 2/3 of hour to wash his dog sport. if he washes him once a week how many hours will brian spend washing sport over 4 weeks
Answer:
2 2/3 hours.
Step-by-step explanation:
If her washs once a week, he will wash 4 times in 4 weeks.
2/3 * 4 = 8/3
2 2/3.
PLEASE HELP DUE IN 5 MINS
The box plots display measures from data collected when 15 athletes were asked how many miles they ran that day.
A box plot uses a number line from 0 to 13 with tick marks every one-half unit. The box extends from 1 to 3.5 on the number line. A line in the box is at 2. The lines outside the box end at 0 and 5. The graph is titled Group A's Miles, and the line is labeled Number of Miles.
A box plot uses a number line from 0 to 13 with tick marks every one-half unit. The box extends from 5 to 10 on the number line. A line in the box is at 7. The lines outside the box end at 0 and 11. The graph is titled Group B's Miles, and the line is labeled Number of Miles.
Which group of athletes ran the least miles based on the data displayed?
Group B, with a median value of 7
Group A, with a median value of 2
Group B, with narrow spread in the data
Group A, with a wide spread in the data
Based on the data displayed in the two box plots, the group of athletes that ran the least miles is Group A, with a median value of 2.
Giving a test to a group of students, the grades and gender are summarized below
The probability that the student was male AND got A is 5/28.
Define probabilityProbability is a measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. A probability of 0.5 (or 50%) indicates that the event is equally likely to occur or not to occur.
There are 56 students total
There are 10 males who got a A.
P(male AND got a A) = (number of males who got a A)/(number total) = (10)/(56) =5/28
Answer in decimal form=0.17
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Use the matrix calculator to solve this linear system for cost per hour of each machine. 100x1 + 130x2 + 16x3 = 3,528 120x1 + 180x2 + 28x3 = 4,864 160x1 + 190x2 + 10x3 = 4,920 x1 = x2 = x3 =
Answer:
x1 = 10
x2 = 16
x3 = 28
Step-by-step explanation:
edge 2023
The cone-shaped paper cup is 2/3 full of sand. What is the volume of the part of the cone that is filled with sand?
F 102.57 cm³
G 153.86 cm³
H 307.72 cm³
J 461.58 cm³
102.57 cm³ is the volume of the part of the cone that is filled with sand.
We can start by using the formula for the volume of a cone:
V = (1/3) × π × r² × h
where V is the volume, r is the radius of the circular base, h is the height, and π is a mathematical constant approximately equal to 3.14. Let's assume that the cone-shaped paper cup has a height of h and a radius of r. Since the cup is 2/3 full of sand, the volume of sand in the cup is 2/3 of the total volume of the cone. Therefore, we can express the volume of sand in terms of the total volume of the cone as:
V sand = (2/3) × V
Substituting the formula for the volume of a cone into the above equation, we get:
V sand = (2/3) × (1/3) × π × r² × h
Simplifying the equation, we get:
V sand = (2/9) × π × r² × h
Therefore, the volume of the part of the cone that is filled with sand is (2/9) × π × r² × h.
Since we do not have the values of r and h, we cannot find the exact volume of the sand. However, we can use the given options to make an educated guess.
Let's try substituting the values of r and h from the given options into the equation and see which option gives us a value close to 2/3 of the total volume of the cone.
Option F: V sand = (2/9) × π × (3.3)² × 4.5 ≈ 102.57 cm³
Option G: V sand = (2/9) × π × (4.4)² × 3.0 ≈ 153.86 cm³
Option H: V sand = (2/9) × π × (5.5)² × 3.0 ≈ 307.72 cm³
Option J: V sand = (2/9) × π × (6.6)² × 2.25 ≈ 461.58 cm³
Option F gives us a value close to 2/3 of the total volume of the cone, so the answer is (F) 102.57 cm³
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The ages of a group of lifeguards are listed. 15, 17, 19, 19, 22, 23, 25, 27, 32, 34 If another age of 46 is added to the data, how would the range be impacted?
The range would increase by 12.
The range would decrease by 12.
The range would stay the same value of 19.
The range would stay the same value of 31.
The range would grow by 12 if the data included a second person who is 46 years old.
The difference between the data set's maximum and smallest values is known as the range.
In the given data set, the minimum value is 15 and the maximum value is 34. So, the range is:
Range = maximum value - minimum value
Range = 34 - 15
Range = 19
If we add another age of 46 to the data set, then the new maximum value would be 46 and the new range would be:
New range = new maximum value - minimum value
New range = 46 - 15
New range = 31
So, the range would increase by 12 if another age of 46 is added to the data.
Therefore, the correct answer is: The range would increase by 12.
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x^2/x^2-16+9x/8x+2x^2
[tex]-15+\frac{25x^2}{8}[/tex]
If you are dealt five cards from a standard deck of 52 cards then find the probability of getting one ten and four kings.
Answer:
The probability of getting one ten from the deck of 52 cards is 4/52, since there are four tens in the deck. After one ten is drawn, there are only 51 cards remaining in the deck.
Step-by-step explanation:
The probability of getting four kings from the remaining 51 cards is (4/51) * (3/50) * (2/49) * (1/48), since there are four kings left in the deck after the ten is drawn, and the probability of drawing a king decreases with each card drawn.
Therefore, the probability of getting one ten and four kings is:
(4/52) * (4/51) * (3/50) * (2/49) * (1/48)
= 0.0000026
This is an extremely low probability, since the chances of getting this specific combination of cards is very low.
A boat heading out to sea starts out at Point A, at a horizontal distance of 1433 feet
from a lighthouse/the shore. From that point, the boat's crew measures the angle of
elevation to the lighthouse's beacon-light from that point to be 15°. At some later
time, the crew measures the angle of elevation from point B to be 6°. Find the
distance from point A to point B. Round your answer to the nearest tenth of a foot if
necessary.
As a result, the distance between points A and B is roughly 630.3 feet as the letters "d" for the distance to the lighthouse from point A and "x" for the distance .
what is trigonometric ratios ?Trigonometric ratios, commonly referred to as trigonometric functions, are mathematical relationships between the ratios of the side lengths of a right triangle and its angles. The first three fundamental trigonometric ratios are: The length of the side opposite the angle to the length of the hypotenuse is known as the sine (sin). The cosine (cos) function measures how long the adjacent side is in relation to the hypotenuse. The length of the side that is opposite the angle to the length of the side that is next to it is referred to as the tangent (tan). The reciprocals of sine, cosine, and tangent, respectively, are cosecant (csc), secant (sec), and cotangent (cot), which are additional trigonometric functions. Trigonometric ratios are employed in a number of disciplines, such as mathematics, physics, engineering, and navigation.
given
Trigonometry can be used to resolve this issue. Let's use the letters "d" for the distance to the lighthouse from point A and "x" for the distance to point B. Next, we have:
tan(15°) = (lighthouse height) / d
tan(6°) is equal to (lighthouse height) / (d + x).
In the first equation, "d" can be solved as follows:
D is equal to (lighthouse height) / tan(15°).
This is what we get when we enter it into the second equation:
tan(6°) is equal to (lighthouse height) / (lighthouse height / tan(15°) + x).
tan(6°) is equal to tan(15°) / (tan(15°) / (lighthouse height) + x/d)
The result is obtained by multiplying both sides by (tan(15°) / (height of lighthouse) + x/d):
Tan(6°) + Tan(15°) / (Lighthouse Height + x/d) = Tan(15°)
We can now determine how to solve for "x"
x is equal to d*(tan(6°)*(height of lighthouse)/tan(15°)-1)
When we enter the values from the issue, we obtain:
D=(lighthouse height)/tan(15°) = 3892.72 feet
630.3 feet are equal to x = 3892.72 * (tan(6°) * (height of lighthouse) / tan(15°) - 1)
As a result, the distance between points A and B is roughly 630.3 feet as the letters "d" for the distance to the lighthouse from point A and "x" for the distance .
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In a rice factory, if each kg of rice needs 1 5 th of total amount of raw paddy, how much amount of raw paddy will be needed to manufacture 3 kg of rice? Total amount of paddy available is 300 kg=
Proportionately, if each kilogram of rice needs ¹/₅ th of the total amount of raw paddy or 60 kg, to manufacture 3 kg of rice, the rice factory needs 180 kg of raw paddy.
How is the quantity determined?The quantity of raw paddy required to manufacture 3 kg of rice can be determined by proportions.
Proportion is the equation of two ratios.
The quantity of raw paddy required by each kg of rice = ¹/₅
The total raw paddy available = 300 kg
¹/₅ of 300 kg = 60 (300 x ¹/₅)
Proportionately, the quantity of raw paddy required for 3 kg of rice = 180 (60 x 3).
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PLEASE HELP AND GIVE A GOOD EXPLANATION!!!!!
The statement that is true about the graph shown is the distribution of the data is symmetrical so the mean and the median are likely within 1000 - 1099 photocopies category (A).
When is the data symmetrical?A graph is symmetrical when it has a line of symmetry, which is a vertical or horizontal line that divides the graph into two equal halves that are mirror images of each other. A graph is symmetrical if it has the same shape on both sides of the line of symmetry.
This happens in the graph presented and as a result other measures such as median and mean are expected to be in the center too.
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A factory manufactures parts for ceiling in fan based on the data shown in the graph below how many parts can factory manufactor in 14 hours
By answering the presented question, we may conclude that The equation number of pieces that may be manufactured in 14 hours is: y = 12(14) = 168 components.
What is equation?An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" asserts that the phrase "2x Plus 3" equals the number "9." The purpose of equation solving is to determine the value or values of the variable(s) that will allow the equation to be true. Equations can be simple or complicated, regular or nonlinear, and include one or more elements. The variable x is raised to the second power in the equation "x2 + 2x - 3 = 0." Lines are utilised in many different areas of mathematics, such as algebra, calculus, and geometry.
The following is the number of pieces that the plant can produce in 14 hours:
168 pieces.
As a result, the equation is as follows:
y = kx.
In where k is a proportionality constant denoting the number of pieces that may be created each hour.
With x = 5, y = 60, and so the constant is given as follows:
k = 60/5
k = 12.
As a result, the equation is:
y = 12x.
The number of pieces that may be manufactured in 14 hours is:
y = 12(14) = 168 components.
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Solve2x^2-6x+5 by completing the square
Find the quotient. 2x − 3 x ÷ 7 x2
Answer: 8/7x
Step-by-step explanation:
2x-3x/7x2
rewrite
2x-3/7x x 2
calculate
2x-6/7x
calculate
solution
8/7x
Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, use a calculator to obtain a decimal approximation for the solution.
e^2x-7 -5 = 27132
The solution set expressed in terms of logarithms is {___}
The Solution set is {___}
The solution set (rounded to four decimal places) is: {x ≈ 5.8986}
Starting from the given equation:
[tex]e^(2x-7) - 5 = 27132[/tex]
Adding 5 to both sides:
[tex]e^(2x-7) = 27137[/tex]
Taking the natural logarithm of both sides:
[tex]ln(e^(2x-7)) = ln(27137)[/tex]
Using the property that ln(e^a) = a:
[tex]2x - 7 = ln(27137)[/tex]
Adding 7 to both sides:
[tex]2x = ln(27137) + 7[/tex]
Dividing by 2:
[tex]x = (ln(27137) + 7)/2[/tex]
Therefore, the solution set expressed in terms of natural logarithms is:
[tex]{x | x = (ln(27137) + 7)/2}[/tex]
Using a calculator to approximate the solution:
x ≈ 5.8986
Therefore, the solution set (rounded to four decimal places) is:
{x ≈ 5.8986}
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Prove the value of the expression (36^5-6^9)(38^9-38^8) is divisible by 30 or 37
Answer:
The given expression is divisible by both 30 and 37
Step-by-step explanation:
First, let's consider the expression (36^5-6^9). We can factor out 6^5 from both terms to get:
(36^5-6^9) = 6^5(6^10-36^3)
Next, let's consider the expression (38^9-38^8). We can factor out 38^8 from both terms to get:
(38^9-38^8) = 38^8(38-1)
Now, we can substitute these factorizations back into the original expression:
(36^5-6^9)(38^9-38^8) = 6^5(6^10-36^3)38^8(38-1)
To show that this expression is divisible by 30, we need to show that it is divisible by both 2 and 3. We can see that 6^5 is divisible by both 2 and 3, so the entire expression is divisible by 2 and 3, and hence divisible by 30.
To show that this expression is divisible by 37, we can use Fermat's Little Theorem, which states that if p is a prime number and a is any integer not divisible by p, then a^(p-1) is congruent to 1 mod p. In this case, p=37 and a=6, so we can write:
6^36 ≡ 1 (mod 37)
Multiplying both sides by 6^10 gives:
6^46 ≡ 6^10 (mod 37)
We can use this congruence to simplify the expression we are interested in:
(36^5-6^9)(38^9-38^8) ≡ (6^10-6^9)(1-38^-1) (mod 37)
Simplifying this expression further gives:
(6^10-6^9)(1-38^-1) ≡ 0 (mod 37)
Therefore, the expression (36^5-6^9)(38^9-38^8) is divisible by both 30 and 37.
Ivan's personal information is:
Age
Time at address
Age of auto
Car payment
Housing costs
Checking and
savings accounts
Finance company
reference
Declared
bankruptcy
61
13 years
3 years
$203
Owns Clear
Both
Major credit cards 5
Ratio of debt to
2%
income
PREVIOUS
No
Never
According to the following table, what is his credit score?
27
Answer:
Step-by-step explanation:
Based on the information provided, we can use the following credit score range and corresponding points:
Excellent: 800-850 (23-27 points)
Very good: 750-799 (18-22 points)
Good: 700-749 (13-17 points)
Fair: 650-699 (8-12 points)
Poor: 600-649 (5-7 points)
Bad: below 600 (0-4 points)
Using this range, we can add up the points for Ivan's information:
Age: 23 points (because he is between 60-64 years old)
Time at address: 5 points (because he has lived at his address for more than 10 years)
Age of auto: 5 points (because he has owned his car for 2-4 years)
Car payment: 13 points (because his monthly car payment is between $200-$299)
Housing costs: 23 points (because he owns his home and has no monthly mortgage or rent payment)
Checking and savings accounts: 5 points (because he has both types of accounts)
Finance company reference: 5 points (because he has a reference from a finance company)
Declared bankruptcy: 0 points (because he has never declared bankruptcy)
Adding up all these points, we get a total of 79 points. Based on the credit score range and points listed above, this falls into the "Excellent" category, which corresponds to a credit score between 800-850. Therefore, Ivan's credit score is likely to be in that range.
Someome help! (identify the solid figures in number order like 1. rectangle 2. square, etc. )
The identification of the solid shapes are;
1. Triangular prism
2. cylinder
3. sphere
4. cone
5. cuboid
6. pyramid
7. pyramid
8. sphere
9. cylinder
10. pyramid
11. cube
12. cube
13. cone
14. cylinder
15. sphere
16. cylinder
What are solid shapes?Solid shapes are three-dimensional (3D) geometric shapes that occupy some space and have length, breadth, and height. Solid shapes are classified into various categories. Some of the shapes have curved surfaces; some of them are in the shape of pyramids or prisms.
Examples of solid shapes include: prisms, pyramids, cone , cylinder, sphere cuboid , cube e.t.c
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The painting shown at the right
has an area of 360 in2. What is
the value of x?
X =
(3x + 2) in.
Required value of x is 20/3 inches.
What is area of the square?
Side × Side
Given, the area of the painting, which is 360 square inches, and we know that the side of the square is (3x + 2) inches. So we can set up an equation,
(3x + 2)² = 360
Expanding the left side,
9x²+ 12x + 4 = 360
Subtracting 360 from both sides,
9x² + 12x - 356 = 0
Now we can use the quadratic formula to solve for x,
[tex]x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} [/tex]
In this case, a = 9, b = 12, and c = -356. Plugging these values into the formula, we get:
[tex]x = \frac{ - 12 \pm \sqrt{ {12}^{2} - 4 \times 9} }{2 \times 9} \\ = \frac{ - 12 \pm \sqrt{144 - 36} }{18} [/tex]
By solving,we will get [tex]x = - \frac{7}{3} \: or \: \frac{20}{3} [/tex]
Since the side length of the square must be positive, we can ignore the negative solution and conclude that the value of x is 20/3 inches.
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Solve the equation. 72n–6 = 1
here you go i think im right
If an interior angle of a regular polygon has a measure of 120 degrees, how many sides does it have?
Step-by-step explanation:
The formula to find the interior angle of a polygon is 180(n - 2) over n.
So, we know that the interior angle of the polygon which is 120, therefore we can produce this equation, 180(n - 2) over 2 n, equals to 120.
Now we use algebric method to solve for n.
180n − 360 = 120n
60n = 360
n = 6
So, the answer is = 6 sideswhat is .00024 as simple fraction
[tex]0.\underline{00024}\implies \cfrac{000024}{1\underline{00000}}\implies \cfrac{24}{100000}\implies \cfrac{8\cdot 3}{8\cdot 12500}\implies \cfrac{3}{12500}[/tex]
Let G := {a1, . . . , an} be a finite abelian group such that n := |G| is odd. Prove that
a1 + · · · + an = 0.
We have shown that the sum of the elements in G is equal to zero, as required.
What is abelian group?An abelian group, also known as a commutative group in mathematics, is a set of elements where the outcome of applying the group operation on two elements of the set does not depend on the order in which the elements are written. The group operation is hence commutative. The integers and real numbers both form abelian groups when addition is used as an operation, and the idea of an abelian group can be seen as a generalisation of these cases.
We can prove this statement using the fact that the sum of the elements in an abelian group is always equal to zero.
Since G is a finite abelian group, every element ai in G has an inverse, denoted by -ai. Furthermore, since G is abelian, the order in which we add the elements does not matter. Therefore, we can rearrange the terms in the sum a1 + a2 + ... + an so that all of the terms are paired with their inverse:
a1 + a2 + ... + an = (a1 + (-a1)) + (a2 + (-a2)) + ... + (an + (-an))
Since n is odd, we have an odd number of elements in G, which means that we have an odd number of pairs of the form ai + (-ai). Therefore, there is exactly one element in the sum that is not paired with its inverse, which is either ai or -ai for some i.
Without loss of generality, suppose that ai is the element that is not paired with its inverse. Then, we have:
a1 + a2 + ... + ai + ... + an = (a1 + (-a1)) + (a2 + (-a2)) + ... + (ai + (-ai)) + ... + (an + (-an))
= 0 + 0 + ... + 0 + ai + (-ai) + ... + 0
= ai - ai
= 0
Therefore, we have shown that the sum of the elements in G is equal to zero, as required.
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