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ISEE Upper Level Quantitative : Cylinders

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

What is the volume of a cylinder with a radius of 6 meters and a height of 11 meters? Use 3.14 for $\pi$ .

Note: The formula for the volume of a cylinder is:

$V=\pi r^2h$

Possible Answers:

114.44m^3

207.24m^3

396m^3

1243.44 m^3

1432.23m^3

Correct answer:

1243.44 m^3

Explanation:

To calculate the volume, you must plug into the formula given in the problem. When you plug in, it should look like this: V=(3.14)(6^2)(11) . Multiply all of these out and you get 1243.44m^3 . The units are cubed because volume is always cubed.

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

The volume of a cylinder whose height is twice the diameter of its base is one cubic yard. Give its radius in inches.

Possible Answers:

$\frac{ 9 \sqrt[3]{ 2 \pi^{2} } } {\pi } \textrm{ in}$

$\frac{ 9 \sqrt[3]{ 2 \pi^{2} } } {4 \pi } \textrm{ in}$

$\frac{ 9 \sqrt[3]{ 2 \pi^{2} } } {2 \pi } \textrm{ in}$

$\frac{ \sqrt[3]{ 2 \pi^{2} } } {2\pi } \textrm{ in}$

$\frac{ 18 \sqrt[3]{ 2 \pi^{2} } } {\pi } \textrm{ in}$

Correct answer:

$\frac{ 18 \sqrt[3]{ 2 \pi^{2} } } {\pi } \textrm{ in}$

Explanation:

The volume of a cylinder with base radius and height is

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

The diameter of this circle is ; its height is twice this, or . Therefore, the formula becomes

$V = \pi r^{2} \cdot 4r = 4\pi r^{3}$

Set this volume equal to one and solve for :

$4\pi r^{3} = 1$

$4\pi r^{3} \div 4\pi = 1 \div 4\pi$

$r ^{3}= \frac{1}{4\pi }$

$r =\sqrt[3]{ \frac{1}{4\pi }} =\frac{ \sqrt[3]{ 1 }} { \sqrt[3]{ 4\pi }} =\frac{ \sqrt[3]{ 1 } \cdot \sqrt[3]{ 2 \pi^{2} } } { \sqrt[3]{ 4\pi } \cdot \sqrt[3]{ 2 \pi^{2} }}=\frac{ \sqrt[3]{ 2 \pi^{2} } } {\sqrt[3]{ 8\pi^{3} }}=\frac{ \sqrt[3]{ 2 \pi^{2} } } {2\pi }$

This is the radius in yards; multiply by 36 to get the radius in inches.

$\frac{ \sqrt[3]{ 2 \pi^{2} } } {2\pi } \cdot 36 = \frac{ 18 \sqrt[3]{ 2 \pi^{2} } } {\pi }$

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Example Question #3 : How To Find The Volume Of A Cylinder

What is the height of a cylinder with a volume of $646.875\pi$ in^3 and a radius of 7.5 ?

Possible Answers:

28125.5

11.5

86.25

56250

22.5

Correct answer:

11.5

Explanation:

Recall that the equation of for the volume of a cylinder is:

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

For our values this is:

$646.875\pi = \pi * 7.5^2 * h= \pi * 56.25 * h$

Solve for :

h = 11.5

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Example Question #4 : How To Find The Volume Of A Cylinder

What is the volume of a cylinder with a height of in. and a radius of in?

Possible Answers:

$204\pi$ in^3

$300\pi$ in^3

$180\pi$ in^3

$720\pi$ in^3

$408\pi$ in^3

Correct answer:

$720\pi$ in^3

Explanation:

This is a rather direct question. Recall that the equation of for the volume of a cylinder is:

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

For our values this is:

$A = \pi * 12^2 * 5=720\pi$

This is the volume of the cylinder.

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Example Question #5 : How To Find The Volume Of A Cylinder

What is the volume of a cylinder with a height of in. and a radius of 2.5 in?

Possible Answers:

$87.5\pi$ in^3

$82.5\pi$ in^3

$42.25\pi$ in^3

$70\pi$ in^3

$35\pi$ in^3

Correct answer:

$87.5\pi$ in^3

Explanation:

This is a rather direct question. Recall that the equation of for the volume of a cylinder is:

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

For our values this is:

$A = \pi * 2.5^2 * 14=87.5\pi$

This is the volume of the cylinder.

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Example Question #51 : Solid Geometry

What is the radius of a cylinder with a volume of $562.5\pi$ in^3 and a height of ?

Possible Answers:

$15\pi$

7.5

56.25

$56.25\pi$

Correct answer:

7.5

Explanation:

Recall that the equation of for the volume of a cylinder is:

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

For our values this is:

$562.5\pi = \pi * r^2 * 10$

Solve for :

56.25=r^2

Using a calculator to calculate $\sqrt{56.25}$ , you will see that r = 7.5

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

What is the surface area of a cylinder of height in., with a radius of in?

Possible Answers:

$152\pi$ in^2

$136\pi$ in^2

$240\pi$ in^2

$120\pi$ in^2

$122\pi$ in^2

Correct answer:

$152\pi$ in^2

Explanation:

Recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. The equation for one base is:

$A = \pi * r^2$

For our problem, this is:

$\pi * 4^2 = \pi * 16$

You need to double this for the two bases:

$2 * 16 * \pi = 32 * \pi$

The area of the "outer face" is a little bit trickier, but it is not impossible. It is actually a rectangle that has the height of the cylinder and a width equal to the circumference of the base; therefore, it is:

$2 * \pi * r * h$

For our problem, this is:

$2 * \pi * 4 * 15 = 120 * \pi$

Therefore, the total surface area is:

$32\pi + 120\pi = 152\pi$

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

What is the surface area of a cylinder having a base of radius in and a height of in?

Possible Answers:

$363\pi$ in^2

$33\pi$ in^2

$99\pi$ in^2

$308\pi$ in^2

$212\pi$ in^2

Correct answer:

$308\pi$ in^2

Explanation:

$A = \pi * r^2$

For our problem, this is:

$\pi * 11^2 = \pi * 121$

You need to double this for the two bases:

$2 * 121 * \pi = 242 * \pi$

$2 * \pi * r * h$

For our problem, this is:

$2 * \pi * 11* 3 = 66 * \pi$

Therefore, the total surface area is:

$242 * \pi + 66 * \pi = 308\pi$

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Example Question #3 : How To Find The Surface Area Of A Cylinder

What is the surface area of a cylinder with a height of in. and a diameter of in?

Possible Answers:

$14\pi$ in^2

$57\pi$ in^2

$22\pi$ in^2

$36\pi$ in^2

$28\pi$ in^2

Correct answer:

$36\pi$ in^2

Explanation:

$A = \pi * r^2$

For our problem, this is:

$\pi * 2^2 = \pi * 4$

You need to double this for the two bases:

$2 * 4 * \pi = 8 * \pi$

$2 * \pi * r * h$

For our problem, this is:

$2 * \pi * 2 * 7 = 28 * \pi$

Therefore, the total surface area is:

$28\pi + 8\pi = 36\pi$

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

The volume of a cylinder with height of is $275\pi$ in^3 . What is its surface area?

Possible Answers:

$66\pi$ in^2

$55\pi$ in^2

$160\pi$ in^2

$82\pi$ in^2

$135\pi$ in^2

Correct answer:

$160\pi$ in^2

Explanation:

To begin, we must solve for the radius of this cylinder. Recall that the equation of for the volume of a cylinder is:

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

For our values this is:

$275\pi = \pi * r^2 * 11$

Solving for , we get:

r^2 = 25

Hence, r = 5

Now, recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. The equation for one base is:

$A = \pi * r^2$

For our problem, this is:

$\pi * 5^2 = \pi * 25$

You need to double this for the two bases:

$2 * 25 * \pi = 50 * \pi$

$2 * \pi * r * h$

For our problem, this is:

$2 * \pi * 5 * 11 = 110 * \pi$

Therefore, the total surface area is:

$50\pi + 110\pi = 160\pi$

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ISEE Upper Level Quantitative : Cylinders

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

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

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

Example Question #3 : How To Find The Volume Of A Cylinder

Example Question #4 : How To Find The Volume Of A Cylinder

Example Question #5 : How To Find The Volume Of A Cylinder

Example Question #51 : Solid Geometry

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

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

Example Question #3 : How To Find The Surface Area Of A Cylinder

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

All ISEE Upper Level Quantitative Resources

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