How to find the length of a radius - ISEE Upper Level (grades 9-12) 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

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?

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

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?

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 $50\pi$

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 $24\pi$

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.

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.)

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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