GMAT Math : Calculating the length of a radius

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Example Questions

Example Question #61 : Geometry

The arc \displaystyle \widehat{AB} of a circle measures \displaystyle 90^{\circ }. The chord of the arc, \displaystyle \overline{AB}, has length \displaystyle x + 2. Give the radius of the circle.

Possible Answers:

\displaystyle x \sqrt{2} + 2 \sqrt{2}

\displaystyle \pi x+ 2 \pi

\displaystyle 2x+4

\displaystyle \frac{1}{2} x \sqrt{2} + \sqrt{2}

\displaystyle 2 \pi x+ 4 \pi

Correct answer:

\displaystyle \frac{1}{2} x \sqrt{2} + \sqrt{2}

Explanation:

A circle can be divided into four congruent arcs that measure

\displaystyle \frac{1}{4} \cdot 360 ^{\circ } = 90 ^{\circ }.

If the four (congruent) chords of the arcs are constructed, they will form a square with sides of length \displaystyle x + 2. The diagonal of a square has length \displaystyle \sqrt{2} times that of a side, which will be

\displaystyle (x + 2) \sqrt{2} = x \sqrt{2} + 2 \sqrt{2}

A diagonal of the square is also a diameter of the circle; the circle will have radius half this length, or

\displaystyle \frac{1}{2} \left ( x \sqrt{2} + 2 \sqrt{2} \right ) =\frac{1}{2} x \sqrt{2} + \sqrt{2}

Example Question #301 : Problem Solving Questions

Two circles in the same plane have the same center. The larger circle has radius 10; the area of the region between the circles is \displaystyle 50 \pi. What is the radius of the smaller circle?

Possible Answers:

\displaystyle 5 \sqrt{6}

\displaystyle 5 \sqrt{2}

\displaystyle 10

\displaystyle 5

\displaystyle 10 \sqrt{2}

Correct answer:

\displaystyle 5 \sqrt{2}

Explanation:

The area of a circle with radius \displaystyle r is \displaystyle A = \pi r^{2}.

Let \displaystyle R be the radius of the smaller circle. Its area is \displaystyle A_{1} = \pi R^{2}. The area of the larger circle is \displaystyle A_{2} = \pi \cdot 10^{2} = 100 \pi. Since the area of the region between the circles is \displaystyle 50 \pi, and is the difference of these areas, we have

\displaystyle 100 \pi - \pi R^{2} = 50 \pi

\displaystyle 100 \pi - \pi R^{2} - 50 \pi + \pi R^{2} = 50 \pi - 50 \pi + \pi R^{2}

\displaystyle 50 \pi = \pi R^{2}

\displaystyle R^{2} = 50

\displaystyle R = \sqrt{50} = \sqrt {25 } \cdot \sqrt{2} = 5 \sqrt{2}

The smaller circle has radius \displaystyle 5 \sqrt{2}.

Example Question #13 : Calculating The Length Of A Radius

\displaystyle 60^{\circ } arc of a circle measures \displaystyle x+ 4. Give the radius of this circle.

Possible Answers:

\displaystyle \frac{6}{\pi}x+\frac{24}{\pi}

\displaystyle 9 \pi x^{2}+ 72 \pi x+ 144 \pi

\displaystyle 6 \pi x+24 \pi

\displaystyle \frac{3}{\pi}x+\frac{12}{\pi}

\displaystyle 3\pi x+12\pi

Correct answer:

\displaystyle \frac{3}{\pi}x+\frac{12}{\pi}

Explanation:

\displaystyle 60^{\circ } arc of a circle is \displaystyle \frac{60}{360 } = \frac{1}{6} of the circle. Since the length of this arc is \displaystyle x+ 4, the circumference is  this, or

\displaystyle C = 6( x+ 4) = 6x+ 24

The radius of a circle is its circumference divided by \displaystyle 2 \pi; therefore, the radius is

\displaystyle r = \frac{C}{2 \pi} = \frac{6x+24}{2\pi} = \frac{3}{\pi}x+\frac{12}{\pi}

Example Question #14 : Calculating The Length Of A Radius

The arc \displaystyle \widehat{AB} of a circle measures \displaystyle 60^{\circ }. The chord of the arc, \displaystyle \overline{AB}, has length \displaystyle x + 2. Give the length of the radius of the circle.

Possible Answers:

\displaystyle x+2

\displaystyle 2 \pi x + 4 \pi

\displaystyle 2x+4

\displaystyle \pi x +2 \pi

Correct answer:

\displaystyle x+2

Explanation:

A circle can be divided into \displaystyle 6 congruent arcs that measure

\displaystyle \frac{1}{6} \cdot 360 ^{\circ } = 60 ^{\circ }.

If the \displaystyle 6 (congruent) chords are constructed, the figure will be a regular hexagon. The radius of this hexagon will be equal to the length of one side - one \displaystyle 60 ^{\circ } chord of the circle; this radius will coincide with the radius of the circle. Therefore, the radius of the circle is the length of chord \displaystyle \overline{AB}, or \displaystyle x+2.

Example Question #15 : Calculating The Length Of A Radius

If a monster truck's wheels have circumference of \displaystyle 31.4 ft,  what is the distance from the ground to the center of the wheel?

Possible Answers:

\displaystyle 3ft

\displaystyle 2.5ft

\displaystyle 5ft

\displaystyle 10ft

Correct answer:

\displaystyle 5ft

Explanation:

If a monster truck's wheels have circumference of \displaystyle 31.4 ft,  what is the distance from the ground to the center of the wheel?

 

This question is asking us to find the radius of a circle. the distance from the outside of the circle to the center is the radius. We are given the circumference, so use the following formula:

\displaystyle Circ=2 \pi r

Then, plug in what we know and solve for r

\displaystyle 31.4=2 \pi r

\displaystyle r=\frac{31.4}{2 \pi}=4.997\approx5ft

Example Question #311 : Gmat Quantitative Reasoning

Two circles in the same plane have the same center. The smaller circle has radius 10; the area of the region between the circles is \displaystyle 50 \pi. What is the radius of the larger circle?

Possible Answers:

\displaystyle 15

\displaystyle 5 \sqrt{2}

\displaystyle 20

\displaystyle 6 \sqrt{5}

\displaystyle 5 \sqrt{6}

Correct answer:

\displaystyle 5 \sqrt{6}

Explanation:

The area of a circle with radius \displaystyle r is \displaystyle A = \pi r^{2}.

Let \displaystyle R be the radius of the larger circle. Its area is \displaystyle A_{1} = \pi R^{2}. The area of the smaller  circle is \displaystyle A_{2} = \pi \cdot 10^{2} = 100 \pi. Since the area of the region between the circles is \displaystyle 50 \pi, and is the difference of these areas, we have

\displaystyle \pi R^{2} - 100 \pi = 50 \pi

\displaystyle \pi R^{2} - 100 \pi+ 100 \pi = 50 \pi + 100 \pi

\displaystyle \pi R^{2} = 150 \pi

\displaystyle R^{2} = 150

\displaystyle R = \sqrt{150} = \sqrt {25 } \cdot \sqrt{6} = 5 \sqrt{6}

The smaller circle has radius \displaystyle 5 \sqrt{6}.

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