ISEE Upper Level Quantitative : Radius

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

Example Question #1 : How To Find The Length Of A Radius

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

Possible Answers:

It cannot be determined from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Correct answer:

(b) is greater.

Explanation:

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

The area of Circle B is \displaystyle 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 \displaystyle 2 (2r ) = 4r.

(a) Twice the radius of circle B is \displaystyle 2 (2r) = 4r.

(b) The sum of the radii of Circles A and B is \displaystyle r + 4r = 5r.

This makes (b) greater.

Example Question #2 : Circles

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

Possible Answers:

\displaystyle 1 \textrm{ ft }3 \textrm{ in }

\displaystyle 1 \textrm{ ft }4 \textrm{ in } 

\displaystyle 2 \textrm{ ft }8 \textrm{ in }

\displaystyle 2 \textrm{ ft }

\displaystyle 2 \textrm{ ft }6 \textrm{ in }

Correct answer:

\displaystyle 1 \textrm{ ft }4 \textrm{ in } 

Explanation:

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 \displaystyle 1 \frac{3}{4} or \displaystyle \frac{7}{4} times this circumference. The length of the minute hand is the radius of this circle \displaystyle r, and the circumference of the circle is \displaystyle C = 2 \pi r, so the distance the tip travels is \displaystyle \frac{7}{4} this, or

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

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

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

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

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

This is equivalent to 1 foot 4 inches.

Example Question #1 : Radius

The tip of the minute hand of a giant clock has traveled \displaystyle 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

Possible Answers:

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(B) is greater

(A) is greater

Correct answer:

(B) is greater

Explanation:

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, 

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

\displaystyle 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 \displaystyle r is the length of the minute hand. We can use the circumference formula to find this:

\displaystyle 2 \pi r = C

\displaystyle 2 \pi r = 16 \pi

\displaystyle 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.

Example Question #141 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Compare the two quantities:

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

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

Possible Answers:

The relationship cannot be determined from the information given.

The quantity in Column A is greater.

 

The two quantities are equal.

The quantity in Column B is greater. 

Correct answer:

The quantity in Column B is greater. 

Explanation:

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

\displaystyle A=\pi * r^2

\displaystyle C=2*\pi*r

For our two quantities, we have:

Quantity A

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

Therefore, \displaystyle r^2 = 64

Taking the square root of both sides, we get: \displaystyle r = 8

Quantity B

\displaystyle 50\pi = 2 * \pi * r

Therefore, \displaystyle r = 25

Therefore, quantity B is greater.

Example Question #1 : Radius

Compare the two quantities:

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

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

 

Possible Answers:

The two quantities are equal.

The quantity in Column B is greater. 

The relationship cannot be determined from the information given.

The quantity in Column A is greater.

Correct answer:

The two quantities are equal.

Explanation:

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

\displaystyle A=\pi * r^2

\displaystyle C=2*\pi*r

For our two quantities, we have:

Quantity A

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

Therefore, \displaystyle r^2 = 144

Taking the square root of both sides, we get: \displaystyle r = 12

Quantity B

\displaystyle 24\pi = 2 * \pi * r

Therefore, \displaystyle r = 12

Therefore, the two quantities are equal.

Example Question #1 : Radius

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

Possible Answers:

\displaystyle 2.5x^{2}+2x+1

\displaystyle 2.5x^{2}+2x-1

\displaystyle 7x-1

\displaystyle 2(x^{2}+x)-1

Correct answer:

\displaystyle 2.5x^{2}+2x-1

Explanation:

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

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

\displaystyle 2.5x^{2}+2x-1

Example Question #3 : Radius

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

(Assume \displaystyle x is positive.)

Possible Answers:

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

\displaystyle 9 \pi x

\displaystyle \frac{9x}{\pi}

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

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

Correct answer:

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

Explanation:

The relation between the area of a circle \displaystyle A and its radius \displaystyle r is given by the formula

\displaystyle A = \pi r^{2}

Since 

\displaystyle A= 81x^{2}:

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

We solve for \displaystyle r:

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

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

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

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

Since \displaystyle r is positive, as is \displaystyle x:

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

Example Question #1 : How To Find The Length Of A Radius

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.

Possible Answers:

\displaystyle 4 \sqrt{13}

\displaystyle 40

\displaystyle 8\sqrt{13}

\displaystyle 64

\displaystyle 20

Correct answer:

\displaystyle 4 \sqrt{13}

Explanation:

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

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

and 

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

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

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

The circles will have areas:

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

 

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

The radius can be calculated now:

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

\displaystyle r^{2} = 208

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

Example Question #1 : Radius

Circle B has a radius \displaystyle \frac{2}{3} as long as that of Circle A.

Which is the greater quantity?

(a) The area of Circle A

(b) Twice the area of Circle B

Possible Answers:

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

(a) is greater.

Explanation:

If we call the radius of Circle A \displaystyle r, then the radius of Circle B is \displaystyle \frac{2}{3} r.

The areas of the circles are:

(a) \displaystyle \pi r^{2}

(b) \displaystyle \pi \left( \frac{2}{3} r \right ) ^{2} = \frac{4}{9} \pi r^{2}

Twice the area of Circle B is

 \displaystyle 2 \cdot \pi \left( \frac{2}{3} r \right ) ^{2} =2 \cdot \frac{4}{9} \pi r^{2} = \frac{8}{9} \pi r^{2} < \pi r^{2},

making (a) the greater number.

Example Question #2 : Radius

Circle 1 is inscribed inside a square. The square is inscribed inside Circle 2.

Which is the greater quantity? 

(a) Twice the area of Circle 1

(b) The area of Circle 2

Possible Answers:

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

If the radius of Circle 1 is \displaystyle r, then the square will have sidelength equal to the diameter of the circle, or \displaystyle 2r. Circle 2 will have as its diameter the length of a diagonal of the square, which by the \displaystyle 45 ^{\circ }-45 ^{\circ }-90^{\circ } Theorem is \displaystyle \sqrt{2} times that, or \displaystyle 2r \sqrt{2}. The radius of Circle 2 will therefore be half that, or \displaystyle r \sqrt{2}.

The area of Circle 1 will be \displaystyle A = \pi r^{2}. The area of Circle 2 will be \displaystyle A = \pi \left ( r\sqrt{2} \right ) ^{2} = \pi \left ( r^{2} \cdot 2 \right ) = 2 \pi r^{2}, twice that of Circle 1.

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