Cylinders - GMAT Quantitative

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Question

What is the surface area of a cylinder with a radius of 7 and a height of 3?

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Answer

All we really need here is to remember the formula for the surface area of a cylinder.

$\dpi{100} \small SA=2\pi r^{2}+2\pi rh=2\pi \left ( 49 \right )+2\pi \left ( 7 \right )\left ( 3 \right )=98\pi +42\pi=140\pi$

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Question

The height of a cylinder is twice the circumference of its base. The radius of the base is 9 inches. What is the surface area of the cylinder?

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Answer

The radius of the base is 9 inches, so its circumference is $2 \pi$ times this, or $2 \pi \cdot 9 = 18 \pi$ inches. The height is twice this, or $36 \pi$ inches.

Substitute $r = 9 , h = 36 \pi$ in the formula for the surface area of the cylinder:

$A = 2 \pi r (r+h)$

$A = 2 \pi \cdot 9 \cdot (9+36\pi ) = 162 \pi + 648 \pi^{2}$ square inches

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Question

Calculate the surface area of the following cylinder.

(Not drawn to scale.)

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Answer

The equation for the surface area of a cylinder is:

$SA=2\pi rh+2\pi r^2$

we plug in our values: to find the surface area

$SA=2\pi (4)(7)+2\pi (4)^2$

$SA=2 \pi (28)+32\pi$

$SA=88\pi$

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Question

Calculate the surface area of the following cylinder.

(Not drawn to scale.)

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Answer

The equation for the surface area of a cylinder is

$SA=2\pi rh+2\pi r^2$

We plug in our values into the equation to find our answer.

Note: we were given the diameter of the cylinder (10), in order to find the radius we had to divide the diameter by two.

$SA=2\pi (5)(8)+2\pi (5)^2$

$SA=2\pi (40)+2\pi (25)$

$SA=130\pi$

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Question

A cylinder has a height of 9 and a radius of 4. What is the total surface area of the cylinder?

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Answer

We are given the height and the radius of the cylinder, which is all we need to calculate its surface area. The total surface area will be the area of the two circles on the bottom and top of the cylinder, added to the surface area of the shaft. If we imagine unfolding the shaft of the cylinder, we can see we will have a rectangle whose height is the same as that of the cylinder and whose width is the circumference of the cylinder. This means our formula for the total surface area of the cylinder will be the following:

$SA=2(\pi r^2)+2\pi rh$

$SA=2\pi (4)^2+2\pi (4)(9)$

$SA=32\pi+72\pi$

$SA=104\pi$

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Question

Grant is making a canister out of sheet metal. The canister will be a right cylinder with a height of mm. The base of the cylinder will have a radius of $\frac{5}{\pi}$ mm. If the canister will have an open top, how many square millimeters of metal does Grant need?

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Answer

This question is looking for the surface area of a cylinder with only 1 base. Our surface area of a cylinder is given by:

$SA=2\pi rh+2\pi r^2$

However, because we only need 1 base, we can change it to:

$SA=2\pi rh+\pi r^2$

We know our radius and height, so simply plug them in and simplify.

$SA=2\pi \left( \frac{5}{\pi}\right)150+\pi\left(\frac{5}{\pi}\right)^2=1500+\frac{25}{\pi}$

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Question

Find the surface area of a cylinder whose height is and radius is .

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Answer

To find the surface area of a cylinder, you must use the following equation.

$SA=2\pi{r}h+2\pi{r^2}$

Thus,

$SA=2\pi(3)(6)+2\pi(3^2)=54\pi$

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Question

A right circular cylinder has bases of radius $\pi ^{x}$ ; its height is $\pi ^{y}$ . Give its surface area.

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Answer

The surface area of a cylinder can be calculated from its radius and height as follows:

$A = \pi r ^{2} + \pi rh$

Setting $r = \pi^{x}$ and $h = \pi ^{x}$ :

$A = \pi \cdot ( \pi^{x}) ^{2} + \pi \cdot \pi^{x} \cdot \pi^{y}$

$A = \pi ^{1 } \cdot \pi^{2x} + \pi ^{1 } \cdot \pi^{x} \cdot \pi^{y}$

$A = \pi ^{1 +2x} + \pi ^{1+x+y}$ or $A = \pi ^{ 2x+1 } + \pi ^{x+y+1}$

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Question

What is the volume of a cone with a radius of 6 and a height of 7?

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Answer

The only tricky part here is remembering the formula for the volume of a cone. If you don't remember the formula for the volume of a cone, you can derive it from the volume of a cylinder. The volume of a cone is simply 1/3 the volume of the cylinder. Then,

$volume = \frac{\pi r^{2}h}{3} = \frac{\pi\cdot 6^{2}\cdot 7}{3} = 84\pi$

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Question

What is the volume of a sphere with a radius of 9?

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Answer

$\dpi{100} \small volume = \frac{4}{3}\pi r^{3} = \frac{4}{3}\pi\times 9^{3} = 972\pi$

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Question

What is the volume of a cylinder that is 12 inches high and has a radius of 6 inches?

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Answer

$volume=\pi r^{2}h=\pi 6^{2}12=432\pi$

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Question

A cylindrical gas tank is 30 meters high and has a radius of 10 meters. How much oil can the tank hold?

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Answer

$V=\pi r^{2}h=\pi (10^{2})30=3000\pi$

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Question

The height and the circumference of a cone are equal. The radius of the cone is 6 inches. Give the volume of the cone.

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Answer

The circumference of a circle with radius 6 inches is $2 \pi \cdot 6 = 12 \pi$ inches, making this the height. The area of the circular base is $B = \pi \cdot 6^{2} = 36 \pi$ square inches. The cone has volume

$V = \frac{1}{3} Bh = \frac{1}{3} \cdot 36 \pi \cdot 12 \pi = 144 \pi ^{2}$ cubic inches.

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Question

The height of a cylinder is twice the circumference of its base. The radius of the base is 10 inches. What is the volume of the cylinder?

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Answer

The radius of the base is 10 inches, so its circumference is $2 \pi$ times this, or $2 \pi \cdot 10 = 20 \pi$ inches. The height is twice this, or $40 \pi$ inches.

Substitute $r = 10 , h = 40 \pi$ in the formula for the volume of the cylinder:

$V = \pi r ^{2} h$

$V = \pi \cdot 10 ^{2}\cdot 40 \pi = 4,000 \pi ^{2}$ cubic inches

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Question

A large cylinder has a height of 5 meters and a radius of 2 meters. What is the volume of the cylinder?

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Answer

We are given the height and radius of the cylinder, which is all we need to calculate its volume. Using the formula for the volume of a cylinder, we plug in the given values to find our solution:

$V=\pi r^2h$

$V=\pi (2)^2(5)$

$V=20\pi$

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Question

Consider the Circle :

(Figure not drawn to scale.)

Suppose Circle is the base of a cylindrical silo that has a height of . What is the volume of the silo in meters cubed?

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Answer

To find the volume of cylinder, use the following equation:

$\small V=\pi r^2h$

In this equation, is the radius of the base and is the height of the cylinder. Plug in the given value for the height of the silo and simplify to get the answer in meters cubed:

$\small \small V=\pi (15:m)^2*30:m=6750 \pi:m^3$

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Question

A given cylinder has a radius of and a height of . What is the volume of the cylinder?

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Answer

The volume of a cylinder with radius and height is defined as $V=\pi r^{2}h$ . Plugging in our given values:

$V=\pi r^{2}h$

$V=\pi (10)^{2}(15)$

$V=\pi (100)(15)$

$V=1500\pi$

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Question

A cylindrical oil drum has a radius of meters and a height of meters. How much oil can the drum hold?

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Answer

Since we are looking to find out how much oil the drum can hold, we need to find the volume of the drum. The volume of a cylinder with radius and height is defined as $V=\pi r^{2}h$ . Plugging in our given values:

$V=\pi r^{2}h$

$V=\pi (2m)^{2}(3m)$

$V=\pi (4m^{2})(3m)$

$V=12\pi m^{3}$

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Question

Daisy has an empty cylindrical water bottle that has a radius of and a height of . How much water can she add to the bottle to fill it up?

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Answer

Since we are looking to find out how much water the bottle can hold, we need to find the volume of the bottle. The volume of a cylinder with radius and height is defined as $V=\pi r^{2}h$ . Plugging in our given values:

$V=\pi (3cm)^{2}(15cm)$

$V=\pi (9cm^{2})(15cm)$

$V=135\pi cm^{3}$

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Question

Find the volume of a cylinder whose height is and radius is .

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Answer

To find the volume of a cylinder, you must use the following equation:

$V=\pi{r^2}h$

Thus,

$V=\pi(2^2)(8)=32\pi$

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