Radius - GMAT Quantitative Reasoning

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Question

On average, Stephanie walks $2\pi$ feet every seconds. If Stephanie walks at her usual pace, how long will it take her to walk around a circular track with a radius of feet, in seconds?

Answer

The length of the track equals the circumference of the circle.

$C = 2\pi r$

$C=2\pi (200)=400\pi$

$Speed = \frac{distance}{time}$

Therefore, $time = \frac{distance}{speed}$ .

$time = 400\pi: ft \div \frac{2\pi : ft}{4: sec} = 400\pi: ft \times\frac{4:sec}{2\pi: feet}$

$time= 400\pi: ft \times\frac{2:sec}{1\pi: feet} = 800:seconds$

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Question

A circle on the coordinate plane has equation

$x ^{2} - 8x + y ^{2} - 10x = 7$

What is its circumference?

Answer

The standard form of the area of a circle with radius and center is

$(x-h)^{2} + (y-k)^{2} = r^{2}$

Once we get the equation in standard form, we can find radius , and multiply it by $2 \pi$ to get the circumference.

Complete the squares:

$\left (\frac{-8}{2} \right )^{2} = 16,\left (\frac{-10}{2} \right )^{2} = 25$

so $x ^{2} - 8x + y ^{2} - 10x = 7$ can be rewritten as follows:

$x ^{2} - 8x + 16 + y ^{2} - 10x + 25 = 7 + 16 + 25$

$\left ( x -4 \right )^{2} + \left ( y -5 \right ) ^{2} = 48$

$r ^{2} = 48$ ,

so $r = \sqrt{48} = \sqrt{16} \cdot \sqrt{3} = 4 \sqrt{3}$

And $C = 2\pi \cdot 4 \sqrt{ 3} = 8 \pi \sqrt{ 3}$

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Question

A circle on the coordinate plane has equation

$x^{2} + y ^{2} = 40$

Which of the following represents its circumference?

Answer

The equation of a circle centered at the origin is

$x^{2} + y ^{2} = r ^{2}$

where is the radius of the circle.

In this equation, $r ^{2} = 40$ , so $r = \sqrt{40}$ ; this simplifies to

$r = \sqrt{40} = \sqrt{4} \cdot \sqrt{10} = 2 \sqrt{10}$

The circumference of a circle is $C = 2 \pi r$ , so substitute $r = 2 \sqrt{10}$ :

$C = 2 \pi r = 2 \pi \cdot 2 \sqrt{10} = 4 \pi \sqrt{10}$

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Question

A circle on the coordinate plane has equation $x^{2} + y ^{2} = 64$ .

What is its circumference?

Answer

The equation of a circle centered at the origin is

$x^{2} + y ^{2} = r ^{2}$ ,

where is the radius of the circle.

In the equation given in the question stem, $r ^{2} = 64$ , so $r = \sqrt{64} = 8$ .

The circumference of a circle is $C = 2 \pi r$ , so substitute :

$C = 2 \pi r = 2 \pi \cdot 8 = 16 \pi$

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Question

Let be concentric circles. Circle has a radius of , and the shortest distance from the edge of circle to the edge of circle is . What is the circumference of circle ?

Answer

Since are concentric circles, they share a common center, like sections of a bulls-eye target. Since the radius of is less than half the distance from the edge of to the edge of , we must have circle is inside of circle . (It's helpful to draw a picture to see what's going on!)

Now we can find the radius of by adding and , which is And the equation for finding circumfrence is $C =2r\pi$ . Plugging in for gives $40\pi$ .

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Question

Consider the Circle :

(Figure not drawn to scale.)

Suppose Circle represents a circular pen for Frank's mules. How many meters of fencing does Frank need to build this pen?

Answer

We need to figure out the length of fencing needed to surround a circular enclosure, or in other words, the circumference of the circle.

Circumference equation:

$C=2 \pi r$

Where is our radius, which is in this case. Plug it in and simplify:

$C=2 \pi *15=30 \pi:m$

And we have our answer!

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Question

If the radius of a circle is , what is its circumference?

Answer

Using the formula for the circumference of a circle, we can plug in the given value for the radius and calculate our solution:

$C=2\pi r=2\pi (5:cm)=10\pi:cm$

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Question

Megan, a civil engineer, is designing a roundabout for the city of Madison. She knows that the distance from the edge of the roundabout to the center must be 25 meters. Help Megan find the circumference of the roundabout.

Answer

Megan, a civil engineer, is designing a roundabout for the city of Madison. She knows that the distance from the edge of the roundabout to the center must be 25 meters. Help Megan find the circumference of the roundabout.

We are asked to find circumference. In order to do so, look at the following formula:

$C=2\pi r$

Where r is our radius and C is our circumference.

We are indirectly told that our radius is 25 meters, plug it in to get our answer:

$C=2\pi 25m=50 \pi m$

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Question

A circle has radius $\pi ^{t}$ . Give its circumference.

Answer

The circumference of a circle is found using the following formula:

$C =2 \pi r$

Set $r= \pi ^{t}$ :

$C =2 \pi \cdot \pi ^{t} = 2 \cdot \pi ^{1} \pi ^{t} = 2 \pi ^{t+1}$

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Question

Consider the Circle :

(Figure not drawn to scale.)

If Circle represents the bottom of a silo, what is the area of the base of the silo in square meters?

Answer

The area of circle is found by the following equation:

$A=\pi r^2$

where is our radius, which is .

So,

$A=\pi*(15:m)^2=225 \pi:m^2$

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Question

A circle on the coordinate plane is defined by the equation $x^{2}+y^{2}=25$ . What is the area of the circle?

Answer

The equation of a circle centered at the origin of the coordinate plane is $x^{2}+y^{2}=r^{2}$ , where is the radius of the circle.

The area of the circle, in turn, is defined by the equation $A=\pi r^{2}$ .

Since we are provided with the equation $x^{2}+y^{2}=25$ , we can deduce that $r^{2}=25$ and that $A=25\pi$ .

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Question

What is the area of a circle with a diameter of ?

Answer

The area of a circle is defined by $A=\pi r^{2}$ , where is the radius of the circle. We are provided with the diameter of the circle, which is twice the length of .

If , then $r=\frac{d}{2}=\frac{20cm}{2}=10cm$

Therefore:

$A=\pi r^{2}$

$A=\pi (10cm)^{2}$

$A=100\pi cm^2$

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Question

For $10, Brandon can order either a 12"-diameter pizza, two 6"-diameter pizzas, or three 4"-diameter pizzas. Which option is the best value, assuming all pizzas are the same thickness?

Answer

For a circle, $Area = \pi r ^{2}$ .

Therefore, the area of the 12" pizza $=6^{2}\pi =36\pi$ .

The area of the two 6" pizzas $=2\times 3^{2}\pi=2\times 9\pi =18\pi$ .

The area of the three 4" pizzas = $3\times 2^{2}\pi=3\times 4\pi =12\pi$ .

The 12" pizza is the best option.

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Question

A circle is inscribed in a square with area 100. What is the area of the circle?

Answer

A square with area 100 would have a side length of 10, which is the diameter of the circle. The area of a circle is $\pi r^{2}$ , so the answer is $\pi * 5^{2} = 25\pi$ .

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Question

A circle on the coordinate plane has equation

$x^{2} + 4x + y ^{2} + 6y = 12$

What is its area?

Answer

The area of a circle is equal to $A = \pi r ^{2}$ , where is the radius.

The standard form of the area of a circle with radius and center is

$(x-h)^{2} + (y-k)^{2} = r^{2}$

Once we get the equation in standard form, we know $r ^{2}$ , which can be multiplied by $\pi$ .

Complete the squares:

$\left (\frac{4}{2} \right )^{2} = 4 ;\left (\frac{6}{2} \right )^{2} = 9$

so $x^{2} + 4x + y ^{2} + 6y = 12$ can be rewritten as follows:

$x^{2} + 4x + 4 + y ^{2} + 6y + 9 = 12 + 4 + 9$

$\left (x + 2 \right ) ^{2} +\left ( y + 3 \right ) ^{2} = 25$

Therefore, $r^{2} = 25$ and $A = \pi r ^{2} = 25 \pi$ .

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Question

Tom has a rope that is 60 feet long. Which of the following is closest to the largest area that Tom could enclose with this rope?

Answer

The largest square you could make would be ${15}'\times {15}'$ with an area of . However, the largest region that can be enclosed will be accomplished with a circle (so you don't lose distance creating the angles). This circle will have a circumference of 60 ft. This gives a radius of

$\frac{60}{2\pi} = \frac{30}{\pi}$

Then the area will be

$\pi \cdot r^2 = \pi \cdot (\frac{30}{\pi})^2 = \frac{900}{\pi} \approx 286$

This is closer to 280 than to 300

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Question

The above figure shows a square inscribed inside a circle. What is the ratio of the area of the circle to that of the square?

Answer

Let be the radius of the circle. Its area is $A _{\bigcirc } = \pi r^{2}$

The diagonal of the square is equal to the diameter of the circle, or . The area of the square is half the product of its (congruent) diagonals: $A_{\square } = \frac{1}{2} \cdot2r \cdot 2r = 2r^{2}$

This makes the ratio of the area of the circle to that of the square $\frac{A _{\bigcirc }}{A_{\square }} = \frac{\pi r^{2} }{2r^{2} } = \frac{\pi }{2 }$ .

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Question

A square has the same area as a circle with a radius of 12 inches. What is the sidelength of that square, in terms of $\pi$ ?

Answer

The area of a circle with radius 12 is

$A = \pi r^{2} = \pi \cdot 12^{2} = 144\pi$ .

This is also the area of the square, so the sidelength of that square is the square root of the area:

$s = \sqrt{A} = \sqrt{144\pi} = \sqrt{144} \cdot \sqrt{\pi}= 12\sqrt{\pi}$

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Question

What is the area of a circle which goes through the points ?

Answer

As can be seen in this diagram, the three points form a right triangle with legs of length 5 and 12.

A circle through these three points circumscribes this right triangle.

An inscribed right, or $90^{\circ }$ , angle intercepts a $2 \times 90 = 180 ^{\circ }$ arc, or a semicircle, making the hypotenuse a diameter of the circle. The diameter of the circle is therefore the hypotenuse of the right triangle, which we can find via the Pythagorean Theorem:

$d = \sqrt{ 5^{2} + 12 ^{2}} = \sqrt{ 25+ 144} = \sqrt{169} = 13$

The radius of the circle is half this, or $\frac{13}{2}$ .

The area of the circle is therefore:

$A = \pi r^{2} = \pi \cdot \left ( \frac{13}{2} \right) ^{2} =\frac{169\pi }{4}$

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Question

A circle on the coordinate plane has equation $x^{2} + y ^{2} = 49$ .

What is its area?

Answer

The equation of a circle centered at the origin is

$x^{2} + y ^{2} = r ^{2}$ ,

where is the radius of the circle.

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

From the equation given in the question stem, we know that $r ^{2} = 49$ , so we can plug this into the area formula:

$A = \pi \cdot 49 = 49 \pi$

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