Twenty-two people have hair 2.5 inches long and twenty-eight people have hair 5.7 inches long. What is the average hair length of all the people detailed? A. 4.29 B. 5.10 C. 4.92 D. 4.57​

Answer: 9" x 10" x 11" = 990 cubic inches.

990 cubic inches / 5 minutes = 198 cubic inches per minute

22" x 23" x 33" = 16698 cubic inches

16698/198 = 84 and 1/3 minutes

1 hour 24 minutes and 20 seconds

To calculate the maxHP values for levels 62-70 by plugging in the appropriate level values into the formula.

One way to extend the maxHP for levels beyond 60 is to use the same formula that was used to calculate the maxXP for levels 1-60, but with the appropriate values plugged in.

The formula for the maxHP for a given level can be approximated as:

maxHP = baseHP x  (1 + level/5)²

where baseHP is the starting HP for level 1 (in this case, it's 330).

To extend the maxHP for levels 61-70, you can simply plug in the appropriate level values into the formula and calculate the corresponding maxHP values.

For example, to find the maxHP for level 61, you would plug in level = 61:

maxHP = 330 x (1 + 61/5)²

maxHP ≈ 2619.9

Similarly, you can calculate the maxHP values for levels 62-70 by plugging in the appropriate level values into the formula.

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Answer:

20 times

Step-by-step explanation:

There are 5 color options that are all the same size.

So orange = 1/5.

(1/5) * 100 = 20 times

The correct statement regarding the proportional relationships is given as follows:

D. The first bucket will be completely filled 11 minutes before the second bucket.

A proportional relationship is a type of relationship between two quantities in which they maintain a constant ratio to each other.

The equation that defines the proportional relationship is given as follows:

y = kx.

In which k is the constant of proportionality, representing the increase in the output variable y when the constant variable x is increased by one.

For the first bucket, the equation is given as follows:

y = 0.4x.

Meaning that 0.4 gallons are inserted each minute, hence the time to insert 2 gallons is given as follows:

2/0.4 = 5 minutes.

For the second bucket, the constant is taken from the graph as follows:

k = 0.5/4

k = 0.125.

Hence the time is given as follows:

2/0.125 = 16 minutes.

Hence the difference is of:

16 - 5 = 11 minutes.

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The distance between the points are matched as;

1. 10. 6

2. 7. 6

3. 7. 1

Using the Pythagorean theorem, we have that the square of the hypotenuse side is equal to the sum of the squares of the other two sides, we have that;

1. Opposite = 8

Adjacent = 7

Substitute the values

x² =8² + 7²

Find the square

x² = 64 + 49

x = √113

x = 10. 6

2. x²= 3² + 7²

Find the square value

x² = 9 + 49

x² = 58

Find the square root

x = 7. 6

3. x² = 5² + 5²

find the value

x² = 50

x = √50

x = 7. 1

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The relative change is approximately -14.9%, indicating a decrease of approximately 14.9% in per person consumption of beef from 2000 to 2020.

To find the absolute and relative change as a percentage in the U.S. per person consumption of beef from 2000 to 2020, we can follow these steps:

Calculating the absolute change:

Absolute change = Final value - Initial value

= 57.7 pounds - 67.8 pounds

= -10.1 pounds

Step 2: Calculate the relative change:

Relative change = (Absolute change / Initial value) * 100

= (-10.1 pounds / 67.8 pounds) * 100

≈ -14.9%

The absolute change is -10.1 pounds, indicating a decrease in per person consumption of beef.

The relative change is approximately -14.9%, indicating a decrease of approximately 14.9% in per person consumption of beef from 2000 to 2020.

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The correct graph of the inequality ( |x| + 3 > 3) is attached accordingly.

Recall that the absolute value (or modulus) of a real number x is its non-negative value regardless of its sign. For example, 5 has an absolute value of 5, and 5 has an absolute value of 5.

A number's absolute value can be conceived of as its distance from zero along the real number line.

So, given  |x| + 3 > 3

Add minus three to both sides

|x| + 3  + (-3) > 3 + (-3)

|x|  > 0

Thus, the absolute value of x is

x > 0 or x < -0

put in the form of inequality, we have

x > 0 or x < 0

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The equation that represents the situation is a = b- 4.

Since the plant grows at a constant rate, we can assume that the height of the plant is increasing linearly with time.

Let's use the data from Week 1 and Week 2 to find the rate of growth for each plant:

For Plant A: Growth rate = (24 cm - 22 cm) / (2 weeks - 1 week) = 2 cm/week

For Plant B: Growth rate = (28 cm - 26 cm) / (2 weeks - 1 week) = 2 cm/week

Since the growth rate is constant, we can use the equation of a line to model the height of each plant over time:

For Plant A: a = 2t + b, where t is the time in weeks and b is the initial height of the plant.

For Plant B: b = 2t + c, where c is the initial height of Plant B.

We can find the values of b and c by substituting the data from Week 1 into these equations:

For Plant A: 22 = 2(1) + b, so b = 20.

For Plant B: 26 = 2(1) + c, so c = 24.

Now we can substitute these values into the equations for Plant A and Plant B:

Plant A: a = 2t + 20

Plant B: b = 2t + 24

To find an equation that gives Plant A's height in terms of Plant B's height, we can solve the equation for t in terms of b:

b = 2t + 24

2t = b - 24

t = (b - 24) / 2

Then we can substitute this expression for t into the equation for Plant A:

a = 2t + 20

a = 2[(b - 24) / 2] + 20

a = b - 24 + 20

a = b - 4

So the equation we were looking for is:

a = b - 4

Therefore, to find Plant A's height in centimeters given Plant B's height in centimeters, we simply subtract 4 from Plant B's height.

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If you have a system of two equations with two unknowns, and the graphs of the two equations do not intersect, the system must have no solution.

option C.

When we have a system of two linear equations in two variables, it can be represented graphically as two lines in a plane. The solution to the system corresponds to the point of intersection of these lines.

If the two lines do not intersect, then they are parallel. In this case, there is no point that satisfies both equations, which means that there is no solution to the system of equations.

In other words, if the two graphs do not intersect, it means that there are no values of the two unknowns that can simultaneously satisfy both equations.

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Answer:

I would go with 900 but I’m not for sure

Step-by-step explanation:

Answer:

[tex]\\[/tex]

Step-by-step explanation:

In the first year, the trainers will be worth 80% of their original value, which is:

[tex]\rm\implies{£100 \times 0.8 = £80}[/tex]

[tex]\\[/tex]

In the second year, they will be worth 80% of their value after the first year:

[tex]\rm\implies{£80 \times 0.8 = \boxed{\boxed{\sf{\:\:\:\green{£64}\:\:\:}}}}[/tex]

[tex]\\[/tex]

[tex]\\[/tex]

[tex]\therefore[/tex] The trainers will be worth £64 in 2 years' time.

Amount of ice cream can the cone hold is, 9.8 inches³

We have to given that;

An ice cream cone has a height of 6 inches and a diameter of 2.5 inches.

Hence, We get;

Radius of cone = 2.5 / 2 = 1.25 inches

Since, We know that;

Volume of cone is,

V = πr²h/3

v = 3.14 × 1.25² × 6 / 3

V = 9.8 inches³

Hence, Amount of ice cream can the cone hold is, 9.8 inches³

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The  Linear function is y equals negative one third times x plus 1

A linear function is a function that has a constant rate of change, meaning that as one variable (usually x) increases or decreases by a certain amount, the other variable (usually y) also changes by a constant amount.

Out of the four given equations, the only one that represents a linear function is the third one:

y equals negative one third times x plus 1

This is because the equation has a constant rate of change of -1/3 for any change in x. For example, if we increase x by 3, y will decrease by 1 (because 3*(-1/3) = -1). Similarly, if we decrease x by 3, y will increase by 1. This constant rate of change is what makes the equation represent a straight line when graphed.

The other three equations do not represent linear functions.

The first equation, x = 3, is not a function at all because there are multiple possible y values for the given x value of 3.

The second equation, y = 3x2, is a quadratic function because it contains an x2 term, which means that the rate of change of y with respect to x is not constant.

The fourth equation, 3x - 4 = 7, is a linear equation but not a function because for any given x, there is only one corresponding value of y. However, when the equation is solved for y, it becomes y = 3x - 3, which represents a linear function with a constant rate of change of 3.

The  linear function is y equals negative one third times x plus 1

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Answer: 350000

Step-by-step explanation: Let the total sale of noodles by the wholesaler be x.

Given: Commission for the wholesaler is 10%

After receiving a 10% commission the company got rupees 315000.

x-10% of x =315000

x - 10\100 = 315000

x-x\10 = 315000

10x - 10\10 =315000

9x = 3150000

x= 3150000\9

x= 350000

The limit as n approaches infinity of ((n+1)⁶-(n-1)⁶)/((n+1)⁵+(n-1)⁵) is 6.

To evaluate the limit as n approaches infinity of the expression ((n+1)⁶-(n-1)⁶)/((n+1)⁵+(n-1)⁵), we can simplify it using algebraic manipulations.

Let's expand the numerator and denominator using the binomial:

Numerator:

((n+1)⁶-(n-1)⁶) = [n⁶ + 6n⁵ + 15n⁴ + 20n³ + 15n² + 6n + 1] - [n⁶ - 6n⁵ + 15n⁴ - 20n³ + 15n² - 6n + 1]

= 12n⁵ + 40n³ + 12n

Denominator:

((n+1)⁵+(n-1)⁵) = [n⁵ + 5n⁴ + 10n³ + 10n² + 5n + 1] + [n⁵ - 5n⁴ + 10n³ - 10n² + 5n - 1]

= 2n⁵ + 20n³ + 2n

Now, let's simplify the expression by canceling out common terms:

((n+1)⁶-(n-1)⁶)/((n+1)⁵+(n-1)⁵) = (12n⁵ + 40n³ + 12n) / (2n⁵ + 20n³ + 2n)

As n approaches infinity, we can see that the highest power of n in the numerator and denominator is n⁵.

We can divide all terms by n⁵ to determine the limit:

lim(n→∞) [(12n⁵ + 40n³ + 12n) / (2n⁵ + 20n³ + 2n)]

= lim(n→∞) [(12 + 40/n² + 12/n⁴) / (2 + 20/n² + 2/n⁴)]

= 12/2

= 6

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The volume of a sphere can be calculated using the formula V = (4/3)πr³, where V is the volume and r is the radius.

Given that the sphere and the cylinder have the same radius and height, we can assume that the radius of both shapes is the same.

Let's denote the radius as 'r'. The volume of the cylinder is given as 50 ft³.

The volume of the cylinder is calculated using the formula V = πr²h, where h is the height.

Since the height of the cylinder is not given, we cannot determine the exact value of 'r' or calculate the volume of the sphere.

However, if we assume that the height of the cylinder is equal to its diameter (which is twice the radius), we can proceed with the calculation.

If the height of the cylinder is equal to 2r, the volume of the cylinder becomes:

50 = πr²(2r)

50 = 2πr³

Dividing both sides of the equation by 2π gives:

25 = πr³

Now, we can substitute this value of πr³ into the volume formula of the sphere:

V = (4/3)πr³

V = (4/3)(25)

V = 100/3

Therefore, if the height of the cylinder is equal to its diameter, the volume of the sphere would be 100/3 ft³. However, please note that this assumption might not necessarily be valid without further information about the relationship between the height and radius of the cylinder.

Answer: The Volume of the sphere is 100/3 cubic feet.

Step-by-step explanation:

It is clearly seen in the given figures, the height is two times the radius. Therefore, h is equal 2r.

We know that,

The volume of the sphere is π x r^2 x h. (r is radius and h is height)

So putting the values of height, h in the above formula, we get:

πxr^2x2r

= 2 * π * r^3.

= 2 * π * r^3 = 50

= π * r^3 = 25

We get, r^3 = 7.957747155

Therefore, r = 1.996472712

Now that we have the measure of the radius, we can find the volume of the sphere!

(4 / 3) * π * r^3

(4/3) * π * (1.996472712)^3

= (4/3) * π * 7.957747155

= (4/3) * π * (25 / π)

= (4/3) * 25

= 100/3 cubic feet.

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1. The best equation for circle A is (x + 4)² + (y + 2)² = 9. Option D

2. The equation of the circle with coordinates (1, 1) (1, 11) is (x - 1)² + (y - 6)² = 25. Option A

1. For the equation of circle A, we can pick out the coordinates we see.

Circle A has its midpoint at (-4, -2) and its edges at (-4, 1) and (-4, -5).

The equation of a circle in a plane is (x - h)² + (y - k)² = r²

the circle has a radius of 3 units (from -2 to 1 or from -2 to -5)

(x - (-4))² + (y - (-2))² = 3²

when we simplify, it becomes (x + 4)² + (y + 2)² = 9

2. The equation of a circle with diameter AB. A (1, 1), B(1, 11) is

h = (x₁ + x₂)/2 ⇒(1 + 1)/2 = 1

k = (y₁ + y₂)/2⇒ (1 + 11)/2 = 6

the center of the circle is at (1, 6).

A and B which are (1, 1) and (1, 11), it's clear that the diameter is 10 units (11 - 1). Thus, the radius is 5 units.

(x - h)² + (y - k)² = r²

Substituting the obtained values we get

(x - 1)² + (y - 6)² = 5²

which becomes

(x - 1)² + (y - 6)² = 25

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Answer: let me explain!

Step-by-step explanation: So this might seem really confusing at first, but it’s actually suuuuuuper easy :P

So, if you think about it, every five seconds the amount of peanuts…well I don’t know how to explain it but let me show you this with numbers.
(4x2)x2x2x2 and so on. The number multiplies by two every five seconds from there. So figure out how many groups of five second there are in 100 seconds by dividing!

It’s 20!

So since 5 happens 20 times, and every time 5 happens, 2 happens, 2 also happens 20 times! (If that makes any sense)

So that answer would be 8x*twenty twos*

Which is…*drumroll please*

8,388,608
I even double checked!

And for the briefcase skeleton thing

She needs to try it 25 times because there are 5 books and each book can be switched with the other 4 times if the first one doesn’t work, therefore you need to multiply and the equation would be 5^2, or in other words, 25.
Don’t forget ur units!

Hope this helps :P

The partial fraction of the given rational expression is [tex]\frac{25}{(x+1(x+2)} =\frac{25A}{3(x-1)} -\frac{25B}{3(x+2)}[/tex]

Given the following equation below:

[tex]\frac{25}{(x+1(x+2)}[/tex]

We are expression as the sum of partial fractions

Let the numerator of both partial expressions be A and B to have:

[tex]\frac{25}{(x+1(x+2)} =\frac{A}{x-1} +\frac{B}{x+2}[/tex]

Simplifying the expression, we will have;

[tex]\frac{25}{(x+1(x+2)} =\frac{A(x+2)+B(x-1)}{x-1(x+2)}[/tex]

Cancelling the denominator:

25 = A(x+2) + B(x-1)


If x - 1 = 0, then x = 1. Substitute to have:

25 = A(1+2)

3A = 25
A = 25/3

If x + 2 = 0, x = -2

25 = B(-3)
B = -25/3

Substitute into the function to have:

[tex]\frac{25}{(x+1(x+2)} =\frac{25A}{3(x-1)} -\frac{25B}{3(x+2)}[/tex]

This gives the partial fraction of the given rational expression.

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Answer:

2553(base 7) = 969(base 10)

Step-by-step explanation:

3 in units place means 3.

5 in the 7's place means 5 × 7 = 35

5 in the 7²'s place means 5 × 7² = 5 × 49 = 245

2 in the 7³'s place means 2 × 7³ = 2 × 343 = 686

3 + 35 + 245 + 686 = 969

An estimated 3,042 people in attendance would prefer relish on a hot dog.

Given that;

For a recent football game of 8,450 in attendance,

And, 150 people were asked for they prefer on a hot dog.

Now, For number of people in attendance would prefer relish on a hot dog, we can use the proportion of people in the sample who preferred relish:

By proportion we get;

x = (54/150) x 8450

x = 54 * 56 1/3

x = 3042

Therefore, An estimated 3,042 people in attendance would prefer relish on a hot dog.

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Answer:

Step-by-step explanation:

No

for it to be a right angle triangle the longest side would have to be the square root of 8 squared plus 11 squared which is 13.6

answer

no Pythagorean theorem doesnt work

steps

8²+11² should equal 17²

BUT

121+64=185

17²=289

The other solution to the equation x³ + 3x² - x = |x-1| is approximately x~ = -2.6.

1. Start by plotting the graphs of both sides of the equation on a graphing calculator.

2. The left side of the equation, x³ + 3x² - x, can be plotted as a curve.

3. The right side of the equation, |x-1|, can be plotted as a V-shaped graph with the vertex at (1, 0).

4. Find the intersection points of the two graphs.

5. By observing the graph, it can be seen that the two given solutions, a~ = -0.7 and b~ = 0.5, are present.

6. Identify the other solution, which is the third intersection point.

7. Determine the approximate x-coordinate of the third intersection point by reading the value from the graph.

8. Round the obtained x-coordinate to the nearest tenth.

9. The other solution is approximately x~ = -2.6.

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The coordinates for the center of the circle are (1/2, 1), and the length of the radius is 2 units. Option C is a correct choice.

To find the coordinates of the center of the circle and the length of the radius, we need to rewrite the equation of the circle in standard form, which is (x - h)² + (y - k)² = r². Here, (h, k) represents the coordinates of the center of the circle, and r represents the radius.

Given equation: x² + y² - x - 2y - 1 = 0

To convert it into standard form, we complete the square for the x and y terms separately.

Rearranging the equation:

x² - x + y² - 2y = 1

Completing the square for x:

(x² - x + 1/4) + (y² - 2y) = 1 + 1/4

Completing the square for y:

(x² - x + 1/4) + (y² - 2y + 1) = 1 + 1/4 + 1

Simplifying:

(x - 1/2)² + (y - 1)² = 9/4

Comparing it with the standard form, we can see that:

Center coordinates: (1/2, 1)

Radius: √(9/4) = 3/2 = 1.5 units

Therefore, the correct answer is C. The coordinates for the center of the circle are (1/2, 1), and the length of the radius is 2 units.
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Question -

A circle is defined by the equation given below. x² + y² - x - 2y - 1 = 0 What are the coordinates for the center of the circle and the length of the radius?

A. (-1,-1), 2 units −1),

B. (- 1/2 ,1), 4 units

C. (1/2 , 1), 2 units

D. (-1/2 ,-1), 4 units

The height of the triangle must increase by 4.29 feet so that the area would increase by 1.2 square feet.

To calculate the area of a triangle, you can use the formula presented as follows:

[tex]\sf Area = \huge \text(\dfrac{1}{2}\huge \text) \times base \times height[/tex]

In which the parameters are given as follows:

Considering the triangle that has a base of 14 feet and a height of 6 feet, the area of the triangle is given as follows:

[tex]\sf A = 0.5 \times 14 \times 6[/tex]

[tex]\sf A = 42 \ square \ feet.[/tex]

An increase of 1.2 square feet would lead to an area of 43.2 square feet, hence the height would be of:

[tex]\sf 4.2h = 43.2[/tex]

[tex]\sf h = \dfrac{43.2}{4.2}[/tex]

[tex]\sf h = 10.29 \ feet[/tex].

Then the increase in the height would be of:

[tex]\sf 10.29 - 6 = 4.29 \ feet[/tex].

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Answer:

The total number of marbles in the bag is 4 + 12 + 8 + 2 = 26.

The probability of not picking a red marble out of the bag is equal to the number of marbles that are not red divided by the total number of marbles in the bag.

The number of marbles that are not red is 4 (blue) + 8 (green) + 2 (yellow) = 14.

Therefore, the probability of not picking a red marble out of the bag is 14/26 or 7/13.

So the probability of not picking a red marble is 7/13.

a) Value of K is Option 3. 98.83%

b) It takes Option 2. 183 bounces for the ball to bounce back to a height of less than 3 inches (1/4 of a foot).

a) To determine k, we can use the formula for the nth term of a geometric sequence:

[tex]a_n[/tex] = [tex]a_1[/tex] * [tex]r^{(n-1)}[/tex]

where a_1 is the initial term, r is the common ratio, and n is the term number.

We know that [tex]a_1[/tex] = 10 feet and [tex]a_5[/tex] = 9.04 feet, so we can set up the equation:

[tex]a_5[/tex] = [tex]a_1[/tex] * [tex]r^{(5-1)}[/tex]

9.04 = 10 * [tex]k^4[/tex]

Solving for k, we get:

k = [tex](9.04/10)^{(1/4)}[/tex] = 0.9883 or 98.83%

Therefore, the answer is (Option 3) 98.83%.

b) We can use the formula for the nth term of a geometric sequence again to find the height of the ball after the nth bounce:

[tex]a_n[/tex] = 10 * [tex]k^{(n-1)}[/tex]

We want to find the smallest n such that a_n < 1/4 foot, or 0.25 feet. Substituting k = 0.9883, we get:

10 * [tex]0.9883^{(n-1)}[/tex] < 0.25

Solving for n, we get:

n > log(0.25/10) / log(0.9883) + 1

n > 183.3

Since n must be a whole number, the smallest n that satisfies this inequality is n = 184. Therefore, it takes 183 bounces for the ball to bounce back to a height of less than 3 inches (Option 2).

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Answer:

Step-by-step explanation:

1. The figure  is a rectangular prism.

Formula: V = lwh

Given: 13 ft x 7ft x 9 ft

Substitute the values to the formula and solve.

V = lwh

V = 13ft(7ft)(9ft)

V = 819 ft³

2. The given figure is a triangular prism. Notice that the base forms a right triangle.

Formula: V = 1/2bhl

Given:

b = 9m

h = 12m

l = 16 m

Substitute the given values to the formula and solve.

V = 1/2(9m)(12m)(16m)

V = 864 m³

3. This is a cylinder

Formula: V = πr²h

Given:

r = 11 yd

h = 26 yd

Solve:

V = π (11yd)²(26yd)

V = 9883.45 yd³

4. The figure is a rectangular prism.

Formula: V = lwh

Given dimensions: 9m x 9m x 13.5m

Solve:

V = 9m(9m)(13.5m)

V = 1093.5 m³

5. The figure is a triangular prism

Formula: V = 1/2bhl

Given:

b = 15yd

h = 13 yd

l = 24 yd

Solve:

V = 1/2 (15yd)(13yd)(24yd)

V = 2340 yd³

Answer:

32) (x+12)(2)+(y)(2)=25

Step-by-step explanation:

(x-a)(2)+(y-b)(2)=r(2)

(x-(-12)(2)+(y-0)(2)=5(2)

(x+12)+(y)=25

Answer(s):

(32) - [tex](x +12)^2 + y ^2 = 25[/tex]

(33) - [tex](x +2)^2 + (y+6 )^2 = 36[/tex]

(34) - [tex](x -8)^2 + (y+1 )^2 = 100[/tex]

(35) - [tex](x -13)^2 + (y+12 )^2 = 1[/tex]

Step-by-step explanation:

To solve these types of problems, it is only a matter of memorizing the equation of a circle and plugging in values appropriately.

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{The Equation of a Circle:}}\\\\(x - h)^2 + (y - k)^2 = r^2\\\text{Where} \ r \ \text{is the radius of the circle and} \ (h,k) \ \text{is the center of the circle.}\end{array}\right}[/tex]

For the following problems, find the equation of a circle with the given information.

(32) - [tex]\text{Center at} \ (-12,0) \ \text{and a radius,} \ r=5[/tex]

(33) - [tex]\text{Center at} \ (-2,-6) \ \text{and a radius,} \ r=6[/tex]

(34) - [tex]\text{Center at} \ (8,-1) \ \text{and a radius,} \ r=10[/tex]

(35) - [tex]\text{Center at} \ (13,-12) \ \text{and a radius,} \ r=1[/tex]

Question #32:

[tex](x - h)^2 + (y - k)^2 = r^2\\\\\Longrightarrow (x - (-12))^2 + (y - 0)^2 = (5)^2\\\\\Longrightarrow \boxed{ (x +12)^2 + y ^2 = 25}[/tex]

Question #33:

[tex](x - h)^2 + (y - k)^2 = r^2\\\\\Longrightarrow (x - (-2))^2 + (y - (-6))^2 = (6)^2\\\\\Longrightarrow \boxed{ (x +2)^2 + (y+6 )^2 = 36}[/tex]

Question #34:

[tex](x - h)^2 + (y - k)^2 = r^2\\\\\Longrightarrow (x - 8)^2 + (y - (-1))^2 = (10)^2\\\\\Longrightarrow \boxed{ (x -8)^2 + (y+1 )^2 = 100}[/tex]

Question #35:

[tex](x - h)^2 + (y - k)^2 = r^2\\\\\Longrightarrow (x - 13)^2 + (y - (-12))^2 = (1)^2\\\\\Longrightarrow \boxed{ (x -13)^2 + (y+12 )^2 = 1}[/tex]

Thus, all problems have been solved.