Cylinders - ISEE Upper Level Quantitative Reasoning

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Question

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

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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.

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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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Question

A cylinder has the following measurements:

Height: 12in
Diameter: 10in

Find the volume.

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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

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.

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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?

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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

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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.

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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

Your family owns a farm with a silo for storing grain. If the silo is 40 feet tall and 15 feet in diameter, what volume of grain can it hold?

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Answer

Your family owns a farm with a silo for storing grain. If the silo is 40 feet tall and 15 feet in diameter, what volume of grain can it hold?

Begin with the formula for volume of a cylinder.

$V=\pi r^2h$

A cylinder is just a circle with height.

So, we know the height is 40 ft, but what is r?

If you said 15, you would be on track to get the problem wrong. That is because the diameter is 15 ft, so our radius is only 7.5 ft.

Plug these in to get our answer:

$V=\pi (7.5ft)^2(40ft)=2250 \pi ft^3$

Our answer should be

$2250 \pi ft^3$

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Question

Let $\pi = 3.14$

Find the volume of a cylinder with a diameter of 12cm and a height of 7cm.

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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 is 12cm. We know that the diameter is two times the radius. Therefore, the radius is 6cm.

We know the height of the cylinder is 7cm.

We know that $\pi = 3.14$ .

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

$V = 3.14 \cdot (6\text{cm})^2 \cdot 7\text{cm}$

$V = 3.14 \cdot 36\text{cm}^2 \cdot 7\text{cm}$

$V = 3.14 \cdot 252\text{cm}^3$

$V = 791.28\text{cm}^3$

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Question

You have a tennis ball canister which is cylindrical. If the radius of the cylinder is 3.5 cm, and the height is 21 cm, what is the volume of the canister?

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Answer

You have a tennis ball canister which is cylindrical. If the radius of the cylinder is 3.5 cm, and the height is 21 cm, what is the volume of the canister?

Begin with the formula for volume of a cylinder:

$V=\pi r^2 h$

Where r is our radius and h is our height.

Note that the formula is just the formula for the area of a circle times the height, because a cylinder is really just a circle with height.

Now, plug in what we know and solve for V

$V=\pi (3.5cm)^2 (21cm)=\pi(12.25cm*21cm)=257.25 \pi cm^3$

So our answer is:

$V=257.25 \pi cm^3$

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Question

Find the volume of a cylinder with a height of 9in and a radius of 8in.

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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 radius of the cylinder is 8in and the height of the cylinder is 9in.

So, we get

$V = \pi \cdot (8\text{in})^2 \cdot 9\text{in}$

$V = \pi \cdot 64\text{in}^2 \cdot 9\text{in}$

$V = \pi \cdot 576\text{in}^3$

$V = 576\pi \text{ in}^3$

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Question

For a cylinder, if the radius of the base is 4, and the height is 10, what is the volume?

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Answer

Write the formula to find the volume of a cylinder.

$V = \pi r^2h$

Substitute the known dimensions.

$V = \pi (4^2)(10)$

Solve for the volume.

$V =160 \pi$

The answer is: $160 \pi$

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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?

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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.

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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

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

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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?

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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?

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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?

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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?

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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

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.

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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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