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

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Question

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

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Answer

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.

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Question

The time is now 1:45 PM. Since noon, the tip of the minute hand of a large clock has moved $\frac{14\pi}{3}$ feet. How long is the minute hand of the clock?

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Answer

Every hour, the tip of the minute hand travels the circumference of a circle. Between noon and 1:45 PM, one and three-fourths hours pass, so the tip travels $1 \frac{3}{4}$ or $\frac{7}{4}$ times this circumference. The length of the minute hand is the radius of this circle , and the circumference of the circle is $C = 2 \pi r$ , so the distance the tip travels is $\frac{7}{4}$ this, or

$\frac{7}{4} C = \frac{7}{4} \cdot 2 \pi r = \frac{7 \pi r }{2}$

Set this equal to $\frac{14\pi}{3}$ feet:

$\frac{7 \pi r }{2} = \frac{14\pi}{3}$

$\frac{7 \pi r }{2} \times \frac{2}{7 \pi}= \frac{14\pi}{3}\times \frac{2}{7 \pi}$

$r= \frac{2}{3}\times \frac{2}{1} = \frac{4}{3} = 1 \frac{1}{3}$ feet.

This is equivalent to 1 foot 4 inches.

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Question

The tip of the minute hand of a giant clock has traveled $40 \pi$ feet since noon. It is now 2:30 PM. Which is the greater quantity?

(A) The length of the minute hand

(B) Three yards

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Answer

Betwen noon and 2:30 PM, the minute hand has made two and one-half revolutions; that is, the tip of minute hand has traveled the circumference of its circle two and one-half times. Therefore,

$C \cdot 2 \frac{1}{2} = 40 \pi$

$C = 40 \pi \div 2 \frac{1}{2} = \frac{40 \pi}{1} \div \frac{5}{2} = \frac{40 \pi}{1} \cdot \frac{2}{5}= \frac{80 \pi}{5} = 16\pi$ feet.

The radius of this circle is the length of the minute hand. We can use the circumference formula to find this:

$2 \pi r = C$

$2 \pi r = 16 \pi$

$r = 16 \pi \div 2 \pi = 8$

The minute hand is eight feet long, which is less than three yards (nine feet), so (B0 is greater.

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Question

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

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Answer

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:

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

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

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Question

Compare the two quantities:

Quantity A: The radius of a circle with area of $64\pi$

Quantity B: The radius of a circle with circumference of $50\pi$

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Answer

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

$A=\pi * r^2$

$C=2\pir$

For our two quantities, we have:

Quantity A

$64\pi = \pi * r^2$

Therefore,

Taking the square root of both sides, we get:

Quantity B

$50\pi = 2 * \pi * r$

Therefore,

Therefore, quantity B is greater.

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Question

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 $24\pi$

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Answer

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

$A=\pi * r^2$

$C=2\pir$

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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Question

The areas of five circles form an arithmetic sequence. The smallest circle has radius 4; the second smallest circle has radius 8. Give the radius of the largest circle.

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Answer

The area of a circle with radius is $A = \pi r^{2}$ . Therefore, the areas of the circles with radii 4 and 8, respectively, are

$A_{1} = \pi \cdot 4^{2} = 16 \pi$

and

$A_{2} = \pi \cdot 8^{2} = 64\pi$

The areas form an arithmetic sequence, the common difference of which is

$64 \pi - 16 \pi = 48 \pi$ .

The circles will have areas:

$16 \pi, 64 \pi, 112 \pi, 160 \pi , 208 \pi$

Since the area of the largest circle is $208 \pi$ , we can find the radius as follows:

The radius can be calculated now:

$\pi r^{2} = 208 \pi$

$r^{2} = 208$

$r = \sqrt{208} = \sqrt{16} \cdot \sqrt{13} = 4 \sqrt{13}$

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Question

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

(Assume is positive.)

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Answer

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 :

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

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Question

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

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Answer

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

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

$36\pi=2r\pi$

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

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

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Question

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

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Answer

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:

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

$r^{2}=18$

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

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Question

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

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Answer

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 .

$24\pi=2r\pi$

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

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

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Question

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

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Answer

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.

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Question

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

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Answer

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:

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Question

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?

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Answer

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.

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Question

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

Tap to reveal answer

Answer

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.

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Question

The time is now 1:45 PM. Since noon, the tip of the minute hand of a large clock has moved $\frac{14\pi}{3}$ feet. How long is the minute hand of the clock?

Tap to reveal answer

Answer

Every hour, the tip of the minute hand travels the circumference of a circle. Between noon and 1:45 PM, one and three-fourths hours pass, so the tip travels $1 \frac{3}{4}$ or $\frac{7}{4}$ times this circumference. The length of the minute hand is the radius of this circle , and the circumference of the circle is $C = 2 \pi r$ , so the distance the tip travels is $\frac{7}{4}$ this, or

$\frac{7}{4} C = \frac{7}{4} \cdot 2 \pi r = \frac{7 \pi r }{2}$

Set this equal to $\frac{14\pi}{3}$ feet:

$\frac{7 \pi r }{2} = \frac{14\pi}{3}$

$\frac{7 \pi r }{2} \times \frac{2}{7 \pi}= \frac{14\pi}{3}\times \frac{2}{7 \pi}$

$r= \frac{2}{3}\times \frac{2}{1} = \frac{4}{3} = 1 \frac{1}{3}$ feet.

This is equivalent to 1 foot 4 inches.

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Question

The tip of the minute hand of a giant clock has traveled $40 \pi$ feet since noon. It is now 2:30 PM. Which is the greater quantity?

(A) The length of the minute hand

(B) Three yards

Tap to reveal answer

Answer

Betwen noon and 2:30 PM, the minute hand has made two and one-half revolutions; that is, the tip of minute hand has traveled the circumference of its circle two and one-half times. Therefore,

$C \cdot 2 \frac{1}{2} = 40 \pi$

$C = 40 \pi \div 2 \frac{1}{2} = \frac{40 \pi}{1} \div \frac{5}{2} = \frac{40 \pi}{1} \cdot \frac{2}{5}= \frac{80 \pi}{5} = 16\pi$ feet.

The radius of this circle is the length of the minute hand. We can use the circumference formula to find this:

$2 \pi r = C$

$2 \pi r = 16 \pi$

$r = 16 \pi \div 2 \pi = 8$

The minute hand is eight feet long, which is less than three yards (nine feet), so (B0 is greater.

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Question

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

Tap to reveal answer

Answer

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:

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

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

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Question

Compare the two quantities:

Quantity A: The radius of a circle with area of $64\pi$

Quantity B: The radius of a circle with circumference of $50\pi$

Tap to reveal answer

Answer

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

$A=\pi * r^2$

$C=2\pir$

For our two quantities, we have:

Quantity A

$64\pi = \pi * r^2$

Therefore,

Taking the square root of both sides, we get:

Quantity B

$50\pi = 2 * \pi * r$

Therefore,

Therefore, quantity B is greater.

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Question

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 $24\pi$

Tap to reveal answer

Answer

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

$A=\pi * r^2$

$C=2\pir$

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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