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Intermediate Geometry : Solid Geometry

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Example Questions

Example Question #11 : Cylinders

How many gallon cans of paint must be purchased in order to put a single coat of paint over the surface of a cylindrical water tank if the tank is 75 feet high and 25 feet in radius, and each gallon can of paint covers 350 square feet?

Assume that there is a side, a top, and a bottom to be painted.

Possible Answers:

Correct answer:

Explanation:

First, use the formula $A = 2\pi r (r+h)$ to find the surface area of the tank in feet.

$A = 2\pi r (r+h) =A = 2\pi \cdot 25 \cdot (25+75) \approx 15,708$

Now divide by 350, remembering to round up.

$15,708 \div 350 \approx 44.9$

45 cans of paint need to be purchased.

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Example Question #1 : How To Find The Volume Of A Cylinder

Given a cylinder with radius of 5cm and height of 10cm, what is the volume of the cylinder?

Possible Answers:

$300\pi\;cm^2$

$500\pi\;cm^2$

$250\pi \; cm^2$

$200\pi\; cm^2$

$100\pi\;cm^2$

Correct answer:

$250\pi \; cm^2$

Explanation:

The volume of a cylinder is given by $\pi r^2\cdot height$

Notice how the formula for the volume is defined as the area of a circle times the lateral height of the cylinder. It is as if we are taking little paper circles and stacking them one-by-one until we fill up the entire container.

Plugging in the numbers we get:

$\pi (5)^2\cdot 10 = 250\pi$ cm^2

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Example Question #1 : How To Find The Volume Of A Cylinder

The volume of a cylinder is $64\pi$ , what is its height?

Possible Answers:

Any positive real number

Any positive rational number

Correct answer:

Any positive real number

Explanation:

Because both the radius and height are unspecified, any real number could be it's height as long as an matching radius is also chosen. There is no restriction on the height or radius being rational.

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Example Question #1 : How To Find The Volume Of A Cylinder

What is the volume of a hollow cylinder with an outer diameter of $8\; in$ , an inner diameter of $2\; in$ and a length of $18\; in$ ?

Possible Answers:

$270\pi \; in^{3}$

$288\pi \; in^{3}$

$300\pi \; in^{3}$

$250\pi \; in^{3}$

$320\pi \; in^{3}$

Correct answer:

$270\pi \; in^{3}$

Explanation:

The general formula for the volume of a hollow cylinder is given by $V=\pi (r_{o}^{2}-r_{i}^{2})l$ where $r_{o}$ is the outer radius, $r_{i}$ is the inner radius, and is the length.

The question gives diameters and we need to convert them to radii by cutting the diameters in half. Remember, d=2r . So the equation to solve becomes:

$V=\pi (4^{2}-1^{2})18$ or $V=270\pi\; in^{3}$

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Example Question #2 : How To Find The Volume Of A Cylinder

Find the volume of the following right cylinder: Cylinder33

Possible Answers:

$14\pi \ in^{3}$

$80\pi \ in^{3}$

$160\pi \ in^{3}$ .

$40\pi \ in^{3}$

$400\pi \ in^{3}$

Correct answer:

$160\pi \ in^{3}$ .

Explanation:

The correct answer is $160\pi \ in^{3}$ .

The formula for volume of a cylinder is

$\pi r^{2} h$

r=4 and h=10

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Example Question #1 : How To Find The Surface Area Of A Cylinder

If a cylinder has a radius, $\small r$ , of 2 inches and a height, $\small h$ , of 5 inches, what is the total surface area of the cylinder?

Possible Answers:

$\small 28\pi$

$\small 36\pi$

$\small 24\pi$

$\small 70\pi$

$\small 18\pi$

Correct answer:

$\small 28\pi$

Explanation:

The total surface area will be equal to the area of the two bases added to the area of the outer surface of the cylinder. If "unwrapped" the area of the outer surface is simply a rectangle with the height of the cylinder and a base equal to the circumference of the cylinder base. We can use these relationships to find a formula for the total area of the cylinder.

$A=2A_{base}+A_{rectangle}$

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

Use the given radius and height to solve for the final area.

$\small 2\pi(2)^{2} + 2\pi (2)(5)$

$\small 8\pi + 20\pi$

$\small 28\pi$

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Example Question #2 : How To Find The Volume Of A Cylinder

A right cylinder has a diameter of $2\:in$ and a height of $10\:in$ . What is the volume of this cylinder?

Possible Answers:

$32\:in^3$

$10 \pi\: in ^3$

$20 \pi\: in^3$

$31.4\:in^2$

$40 \pi\: in^3$

Correct answer:

$10 \pi\: in ^3$

Explanation:

The formula to find the volume of a cylinder is: $V = \pi \cdot r^2 \cdot h$ , where is the radius of the cylinder and is the height of the cylinder.

A good point to start in this kind of a formula-based problem is to ask "What information do I have?" and "What information is missing that I need?"

In this case, the problem provides us with the height of the cylinder and its diameter. We have the component of the equation, but we're missing the component. Can we find out ? The answer is yes! Radius is half of diameter. So in this case, because the diameter is $2\:in$ , the radius must be $1\:in$ .

Now that we have and , we are ready to solve for the volume after substituting in those values.

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

$V = \pi \cdot (1)^2 \cdot 10$

$V = \pi \cdot \1 \cdot 10$

$V = 10 \pi\: in^3$

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Example Question #7 : How To Find The Volume Of A Cylinder

You have just bought a farm that includes a storage silo. You want to find the volume for the 20 foot tall silo in the shape of a cylinder. You measure the circumference of the silo, it is listed below.

$Circumference=10\pi\ ft$

Leave answer in terms of $\pi$ .

Possible Answers:

$500\pi\ ft^{3}$

$25\pi\ ft^{3}$

$2000\pi\ ft^{3}$

$166\frac{2}{3}\pi\ ft^{3}$

Correct answer:

$500\pi\ ft^{3}$

Explanation:

To find the volume of a cylinder we need to find the area of the base and then multiply that by the height. The base is in the shape of a circle so the formula is given below:

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

To find the radius of the base, we use what was found from the measurement for circumference of the base. We set the formula for circumference equal to the measured circumference given in the problem.

$2\pi\ (r)=10\pi$

From this we can solve to find that the radius =5. Which we can plug into the original Volume formula. With the given height of 20 ft.

$V=\pi\ 5^{2}\ (20)$

Simplifying will give us the volume of the cylinder.

$V=500\pi\ ft^{3}$

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Example Question #26 : Cylinders

You are painting a water tank black. You are trying to find the surface area of the tank so you measure the height and diameter of the tank. The height is 2.5 meters, and diameter is 4 meters. What is the total surface area of the tank?

Possible Answers:

$10 \pi \; m^2$

$18 \pi \; m^2$

$42 \pi \; m^2$

$14 \pi \; m^2$

$8 \pi \; m^2$

Correct answer:

$18 \pi \; m^2$

Explanation:

To find the surface area of a cylinder we need to find the area of the circular top and bottom and the area of the rectangular (but rounded) side, then add them together. Lets start by finding the area of the circular ends to the cylinder, remember that we measured the diameter (d) and we need the radius (r) to find the area of a circle, the radius is half of the diameter:

$A = r^2\pi = 2^2 \pi = 4\pi \; m^2$ (area of one circle)

Don't forget that we have two circles on the cylinder that have this area:

$4\pi \; m^2 + 4\pi \; m^2 = 8\pi \; m^2$ (area for both circles)

Now we just need to find the area of rectangular side of the cylinder. To do this we will take the hieght of the cylinder, 2.5 m, and multiply by the circumference (C) of the circle (the width of the rectangular side of the cylinder).

$C = 2\pi r \; \text{or} \; C= d\pi = 4\pi \; cm$

So now that we know the circumference we can multiply by the cylinder height to find the area of the rectangular side:

$4\pi \; cm \times 2.5 \; cm = 10\pi \; cm^2$ (area for rectangular side)

Now just add to find the total surface area:

$8\pi \; cm^2 + 10\pi \; cm^2 = 18\pi \; cm^2$

and that is our total surface area for the cylinder!

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Example Question #27 : Cylinders

A can of tomato soup has a diameter of 3.5 inches and a height of 8 inches. What is the volume of the can of soup?

Possible Answers:

$14\pi \; cm^3$

$24.5\pi \; cm^3$

$12.25\pi \; cm^3$

$3.0625\pi \; cm^3$

$28\pi \; cm^3$

Correct answer:

$24.5\pi \; cm^3$

Explanation:

To find the volume of a soup can, or a cylinder, we use the formula for volume:

$\text {Base Area x Height = Volume}$

The base of our cylinder is a circle and to find the area of the circle we use:

$\text {Area =} \; \pi\times r^2 \;, \text {here our radius = }\frac{3.5}{2} = 1.75cm$

Now we can use the formula for the area of our circular base:

$\text {Area =} \; \pi\times (1.75)^2 = 3.0625\pi \; cm^2$

All we need to do now is multiply the base by the hieght of our cylinder!

$3.0625 \pi \; cm^2 \times 8cm = 24.5\pi \; cm^3$

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Intermediate Geometry : Solid Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #11 : Cylinders

Example Question #1 : How To Find The Volume Of A Cylinder

Example Question #1 : How To Find The Volume Of A Cylinder

Example Question #1 : How To Find The Volume Of A Cylinder

Example Question #2 : How To Find The Volume Of A Cylinder

Example Question #1 : How To Find The Surface Area Of A Cylinder

Example Question #2 : How To Find The Volume Of A Cylinder

Example Question #7 : How To Find The Volume Of A Cylinder

Example Question #26 : Cylinders

Example Question #27 : Cylinders

All Intermediate Geometry Resources

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