Cylinders - ISEE Upper Level Quantitative Reasoning

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Question

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

Answer

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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Question

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

Answer

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:

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

You need to double this for the two bases:

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

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:

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

Therefore, the total surface area is:

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

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Question

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

Answer

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. Notice, however that the diameter is inches. This means that the radius is . Now, the equation for one base is:

$A = \pi * r^2$

For our problem, this is:

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

You need to double this for the two bases:

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

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:

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

Therefore, the total surface area is:

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

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Question

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

Answer

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:

Hence,

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:

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

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:

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

Therefore, the total surface area is:

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

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Question

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

Answer

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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Question

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:

$V=\pi r^2h$

Answer

To calculate the volume, you must plug into the formula given in the problem. When you plug in, it should look like this: . Multiply all of these out and you get . The units are cubed because volume is always cubed.

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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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 :

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

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Question

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

Answer

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 :

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Question

A cylinder has the following measurements:

Height: 12in
Diameter: 10in

Find the volume.

Answer

To find the volume of a cylinder, we will use the following formula:

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

where r is the radius, and h is the height of the cylinder.

Now, we know the diameter of the cylinder is 10in. We also know the diameter is two times the radius. Therefore, the radius is 5in.

We know the height of the cylinder is 12in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot (5\text{in})^2 \cdot 12\text{in}$

$V = \pi \cdot 25\text{in}^2 \cdot 12\text{in}$

$V = \pi \cdot 300\text{in}^3$

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Question

If a cylinder has a height of 6 mm and a radius of 12 mm, what is its surface area?

Answer

If a cylinder has a height of 6 mm and a radius of 12 mm, what is its surface area?

Find surface area with the following formula:

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

This works because we are adding the area of the two bases to the area of the side.

Plug in and simplify:

$A=2\pi (12mm)(6mm) +2\pi (12mm)^2=144\pi mm^2+288\pi mm^2=432 \pi mm^2$

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Question

Find the volume of a cylinder with a diameter of 14cm and a height of 8cm.

Answer

To find the volume of a cylinder, we will use the following formula:

$V = \pi r^2 h$

where r is the radius and h is the height of the cylinder.

Now, we know the diameter of the cylinder is 14cm. We also know that the diameter is two times the radius. Therefore, the radius is 7cm.

We know the height of the cylinder is 8cm.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot (7\text{cm})^2 \cdot 8\text{cm}$

$V = \pi \cdot 49\text{cm}^2 \cdot 8\text{cm}$

$V = \pi \cdot 392\text{cm}^3$

$V = 392\pi \text{ cm}^3$

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Question

The axle for a toy car has a length of 4 inches and a diameter of a quarter inch. What is the volume of the axle? Assume it is a cylinder.

Answer

The axle for a toy car has a length of 4 inches and a diameter of a quarter inch. What is the volume of the axle? Assume it is a cylinder.

Use the following formula for volume of a cylinder

$V_{cylinder}=\pi r^2h$

Where r and h are our radius and height, respectively.

In this case, we first need to change our diameter to radius. Because our diameter is one quarter of an inch, our radius will be one eighth of an inch.

Plug it in to get:

$V_{cylinder}=\pi (\frac{1}{8})^24=\pi\frac{1}{64}in^2*4in=\frac{4\pi}{64}in^3$

Simplify to get:

$\frac{4\pi}{64}in^3=\frac{\pi}{16}in^3$

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Question

You are visiting the drive-through at the bank. You put you money in a cylindrical tube with a height of 8 inches and a radius of 2 inches. What is the volume of the tube?

Answer

You are visiting the drive-through at the bank. You put you money in a cylindrical tube with a height of 8 inches and a radius of 2 inches. What is the volume of the tube?

Begin with the formula for volume of a cylinder:

$V_{cylinder}=\pi r^2h$

We have r and h, which are our radius and height, respectively. Plug them in and solve

$V_{cylinder}=\pi (2in)^28in=\pi32in^2$

Making our answer:

$32\pi in^2$

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Question

Find the volume of a cylinder with the following measurements:

diameter = 8cm

height = 5cm

Answer

To find the volume of a cylinder, we will use the following formula:

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

where r is the radius and h is the height of the cylinder.

Now, we know the diameter of the cylinder is 8cm. We know that the diameter is two times the radius. Therefore, the radius is 4cm.

We also know the height of the cylinder is 5cm.

Knowing this, we can substitute into the formula.

$V = \pi \cdot (4\text{cm})^2 \cdot 5\text{cm}$

$V = \pi \cdot 16\text{cm}^2 \cdot 5\text{cm}$

$V = \pi \cdot 80\text{cm}^3$

$V = 80\pi \text{ cm}^3$

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Question

Find the volume of a cylinder with a diameter of 8in and a height of 7in.

Answer

To find the volume of a cylinder, we will use the following formula:

$V = \pi r^2 h$

where r is the radius and h is the height of the cylinder.

Now, we know the diameter of the cylinder is 8in. We also know the diameter is two times the radius. Therefore, the radius is 4in.

We also know the height of the cylinder is 7in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot(4\text{in})^2 \cdot 7\text{in}$

$V = \pi \cdot 16\text{in}^2 \cdot 7\text{in}$

$V = \pi \cdot 112\text{in}^3$

$V = 112\pi \text{ in}^3$

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Question

Find the volume of a cylinder with a diameter of 14in and a height of 12in.

Answer

To find the volume of a cylinder, we will use the following formula:

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

where r is the radius, and h is the height of the cylinder.

Now, we know the diameter of the cylinder is 14in. We also know the diameter is two times the radius. Therefore, the radius is 7in.

We know the height of the cylinder is 12in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot (7\text{in})^2 \cdot 12\text{in}$

$V = \pi \cdot 49\text{in}^2 \cdot 12\text{in}$

$V = \pi \cdot 588\text{in}^3$

$V = 588\pi \text{ in}^3$

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Question

A cylinder has the following measurements:

Height: 6in
Diameter: 16in

Find the volume.

Answer

To find the volume of a cylinder, we will use the following formula:

$V = \pi r^2 h$

where r is the radius and h is the height of the cylinder.

Now, we know the diameter of the cylinder is 16in. We also know the diameter is two times the radius. Therefore, the radius is 8in.

We also know the height of the cylinder is 6in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot(8\text{in})^2 \cdot 6\text{in}$

$V = \pi \cdot 64\text{in}^2 \cdot 6\text{in}$

$V = \pi \cdot 384\text{in}^3$

$V = 384\pi \text{ in}^3$

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