How to find the length of a radius

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ISEE Upper Level Quantitative Reasoning › How to find the length of a radius

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You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be $169 \pi m^2$ .

What is the radius of the crater?

CORRECT

Cannot be determined from the information provided

Explanation

You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be $169 \pi m^2$ .

What is the radius of the crater?

To solve this, we need to recall the formula for the area of a circle.

$A=\pi r^2$

Now, we know A, so we just need to plug in and solve for r!

$169 \pi m^2=\pi r ^2$

Begin by dividing out the pi

Then, square root both sides.

$r=\sqrt{169m^2}=13m$

So our answer is 13m.

The area of Circle B is four times that of Circle A. The area of Circle C is four times that of Circle B. Which is the greater quantity?

(a) Twice the radius of Circle B

(b) The sum of the radius of Circle A and the radius of Circle C

(b) is greater.

CORRECT

(a) is greater.

(a) and (b) are equal.

It cannot be determined from the information given.

Explanation

Let be the radius of Circle A. Then its area is $\pi r^{2}$ .

The area of Circle B is $4\pi r^{2} =2^{2}\pi r^{2} = \pi (2r)^{2}$ , so the radius of Circle B is twice that of Circle A; by a similar argument, the radius of Circle C is twice that of Circle B, or .

(a) Twice the radius of circle B is .

(b) The sum of the radii of Circles A and B is .

This makes (b) greater.

Inscribed angle

Refer to the above diagram. $\overarc {ACB}$ has length $60 \pi$ . Give the radius of the circle.

CORRECT

Explanation

Inscribed $\angle ACB$ , which measures $72 ^{\circ }$ , intercepts a minor arc with twice its measure. That arc is $\overarc {AB}$ , which consequently has measure

$72 ^{\circ } \times 2 = 144 ^{\circ }$ .

The corresponding major arc, $\overarc {ACB}$ , has as its measure

$360 ^{\circ } - 144 ^{\circ } = 216 ^{\circ }$ , and is

$\frac{ 216 }{360 } = \frac{ 216 \div 72 }{360 \div 72 } = \frac{3}{5}$

of the circle.

If we let be the circumference and be the radius, then $\overarc {ACB}$ has length

$\frac{3}{5} C = \frac{3}{5} \cdot 2 \pi r = \frac{6 \pi}{5} r$ .

This is equal to $60 \pi$ , so we can solve for in the equation

$\frac{6}{5} \pi r = 60 \pi$

$\frac{5}{6 \pi } \cdot \frac{6 \pi }{5} r = \frac{5}{6 \pi } \cdot 60 \pi$

The radius of the circle is 50.

A circle has a circumference of $768\pi m$ . What is the radius of the circle?

CORRECT

Not enough information to determine.

Explanation

A circle has a circumference of $768\pi m$ . What is the radius of the circle?

Begin with the formula for circumference of a circle:

$Circumference=2\pi r$

Now, plug in our known and work backwards:

$768\pi m=2\pi r$

Divide both sides by two pi to get:

The area of a circle is $81x^{2}$ . Give its radius in terms of .

(Assume is positive.)

$\frac{9x}{\sqrt{\pi}}$

CORRECT

$\frac{9x}{\pi}$

$\frac{81x^{2}}{\pi}$

$\frac{81x^{2}}{2\pi}$

$9 \pi x$

Explanation

The relation between the area of a circle and its radius is given by the formula

$A = \pi r^{2}$

Since

$A= 81x^{2}$ :

$81x^{2} = \pi r^{2}$

We solve for :

$\pi r^{2} = 81x^{2}$

$\frac{\pi r^{2} }{\pi}= \frac{81x^{2}}{\pi}$

$r^{2} = \frac{81x^{2}}{\pi}$

$\sqrt{r^{2} }= \sqrt{\frac{81x^{2}}{\pi}}$

Since is positive, as is :

$r =\frac{ \sqrt{81} \cdot \sqrt{x^{2}}}{\sqrt{\pi}} = \frac{9x}{\sqrt{\pi}}$

If the diameter of a circle is equal to $5x^{2}+4x-2$ , then what is the value of the radius?

$2.5x^{2}+2x-1$

CORRECT

$2.5x^{2}+2x+1$

$2(x^{2}+x)-1$

Explanation

Given that the radius is equal to half the diameter, the value of the radius would be equal to $5x^{2}+4x-2$ divided by 2. This gives us:

$\frac{5x^{2}+4x-2}{2}$

$2.5x^{2}+2x-1$

What is the radius of a circle with circumference equal to $36\pi$ ?

CORRECT

Explanation

The circumference of a circle can be found using the following equation:

$C=d\pi=2r\pi$

$36\pi=2r\pi$

$\frac{36\pi}{\pi}=\frac{2r\pi}{\pi}$

$\frac{36}{2}=\frac{2r}{2}$

What is the value of the radius of a circle if the area is equal to $18\pi$ ?

$3\sqrt{2}$

CORRECT

$2\sqrt{3}$

Explanation

The equation for finding the area of a circle is $\pi r^{2}$ .

Therefore, the equation for finding the value of the radius in the circle with an area of $18\pi$ is:

$\pi r^{2}=18\pi$

$r^{2}=18$

$r=\sqrt{18}=\sqrt{9\cdot 2}=3\sqrt{2}$

What is the radius of a circle with a circumference of $24\pi$ ?

CORRECT

Explanation

The circumference of a circle can be found using the following equation:

$C=2r\pi$

We plug in the circumference given, $24\pi$ into and use algebraic operations to solve for .

$24\pi=2r\pi$

$\frac{24\pi }{\pi }=\frac{2r\pi }{\pi }$

$\frac{24}{2}=\frac{2r}{2}$

Compare the two quantities:

Quantity A: The radius of a circle with area of $144\pi$

Quantity B: The radius of a circle with circumference of $24\pi$

The two quantities are equal.

CORRECT

The quantity in Column A is greater.

The quantity in Column B is greater.

The relationship cannot be determined from the information given.

Explanation

Recall for this question that the formulae for the area and circumference of a circle are, respectively:

$A=\pi * r^2$

$C=2*\pi*r$

For our two quantities, we have:

Quantity A

$144\pi = \pi * r^2$

Therefore,

Taking the square root of both sides, we get:

Quantity B

$24\pi = 2 * \pi * r$

Therefore,

Therefore, the two quantities are equal.

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