Rectangular Solids & Cylinders - GMAT Quantitative

Card 0 of 960

Question

Calculate the surface area of the following cylinder.

(Not drawn to scale.)

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.)

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?

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?

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 .

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.

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?

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?

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?

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?

Answer

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

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Question

Find the length of the edge of a cube given that the volume is .

Answer

To find side length, you must use the equation for volume of a cube and solve for .

$V=s^3\Rightarrow s=\sqrt[3]{V}$

Thus,

$s=\sqrt[3]{64}=4$

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Question

What is the surface area of a box that is 3 feet long, 2 feet wide, and 4 feet high?

Answer

$\dpi{100} \small SA = 2lw + 2lh + 2wh = 2\times 3\times 2 + 2 \times 3\times 4 + 2\times 2\times 4 = 52$

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Question

What is the surface area of a cube with side length 4?

Answer

$SA=6\ast side^{2}=6\ast 4^{2}=6\ast 16=96$

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Question

What is the length of the diagonal of a cube if its side length is ?

Answer

The diagonal of a cube extends from one of its corners diagonally through the cube to the opposite corner, so it can be thought of as the hypotenuse of a right triangle formed by the height of the cube and the diagonal of its base. First we must find the diagonal of the base, which will be the same as the diagonal of any face of the cube, by applying the Pythagorean theorem:

$c^2=18\rightarrow c=\sqrt{18}=3\sqrt{2}$

Now that we know the length of the diagonal of any face on the cube, we can use the Pythagorean theorem again with this length and the height of the cube, whose hypotenuse is the length of the diagonal for the cube:

$(3\sqrt{2})^2+3^2=d^2$

$d^2=27\rightarrow d=\sqrt{27}=3\sqrt{3}$

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Question

is a cube and face has an area of . What is the length of diagonal of the cube ?

Answer

To find the diagonal of a cube we can apply the formula $d=e\sqrt{3}$ , where is the length of the diagonal and where is the length of an edge of the cube.

Since we are given an area of a face of the cube, we can find the length of an edge simply by taking its square root.

$9=s^2 \rightarrow \sqrt 9 =s \rightarrow 3=s$

Here the length of an edge is 3.

Thefore the final andwer is $d=e\sqrt3=3\sqrt{3}$ .

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Question

What is the length of the diagonal of cube , knowing that face has diagonal equal to ?

Answer

To find the length of the diagonal of the cube, we can apply the formula, however, we firstly need to find the length of an edge, by applying the formula for the diagonal of the square.

$d=s\sqrt{2}$ where is the diagonal of face ABCD, and , the length of one of the side of this square.

The length of must be $\sqrt{2}$ , which is the length of the edges of the square.

Therefore we can now use the formula for the length of the diagonal of the cube:

$e\sqrt{3}$ , where is the length of an edge.

Since $e=\sqrt{2}$ , we get the final answer $\sqrt{6}$ .

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Question

If a cube has a side length of , what is the length of its diagonal?

Answer

The diagonal of a cube is the hypotenuse of a right triangle whose height is one side and whose base is the diagonal of one of the faces. First we must use the Pythagorean theorem to find the length of the diagonal of one of the faces, and then we use the theorem again with this value and length of one side of the cube to find the length of its diagonal:

$c^2=32\rightarrow c=\sqrt{32}$

So this is the length of the diagonal of one of the faces, which we plug into the Pythagorean theorem with the length of one side to find the length of the diagonal for the cube:

$4^2+(\sqrt{32})^2=d^2$

$d^2=48\rightarrow d=\sqrt{48}=\sqrt{3*16}=4\sqrt{3}$

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Question

A given cube has an edge length of . What is the length of the diagonal of the cube?

Answer

The diagonal of a cube with an edge length can be defined by the equation $d=s\sqrt{3}$ . Given in this instance, $d=5\sqrt{3}$ .

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Question

A given cube has an edge length of . What is the length of the diagonal of the cube?

Answer

The diagonal of a cube with an edge length can be defined by the equation $d=s\sqrt{3}$ . Given in this instance, $d=9\sqrt{3}$ .

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Question

A given cube has an edge length of $\sqrt{3}$ . What is the length of the diagonal of the cube?

Answer

The diagonal of a cube with an edge length can be defined by the equation $d=s\sqrt{3}$ . Given $s=\sqrt{3}$ in this instance, $d=\sqrt{3}\left ( \sqrt{3} \right )=3$ .

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