ISEE Upper Level Quantitative Reasoning - How to find the volume of a cone

Find the volume of a cone with the following measurements:

Diameter: 14in
Height: 9in

$147\pi \text{ in}^3$

$126\pi \text{ in}^3$

$441\pi \text{ in}^3$

$63\pi \text{ in}^3$

$252\pi \text{ in}^3$

Explanation

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

$V = \pi r^2 \frac{h}{3}$

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

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

We know the height of the cone is 9in.

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

$V = \pi \cdot (7\text{in})^2 \cdot \frac{9\text{in}}{3}$

$V = \pi \cdot 49\text{in}^2 \cdot 3\text{in}$

$V = \pi \cdot 147\text{in}^3$

$V = 147\pi \text{ in}^3$

A cone has height 18 inches; its base has radius 4 inches. Give its volume in cubic feet (leave in terms of $\pi$ )

$\frac{1}{18} \pi \textrm{ ft}^{3}$

$\frac{1}{6} \pi \textrm{ ft}^{3}$

$\frac{1}{3} \pi \textrm{ ft}^{3}$

$\frac{1}{12} \pi \textrm{ ft}^{3}$

$\frac{1}{4} \pi \textrm{ ft}^{3}$

Explanation

Convert radius and height from inches to feet by dividing by 12:

Height: 18 inches = $18 \div 12 = \frac{18}{12} = \frac{3}{2}$ feet

Radius: 4 inches = $4 \div 12 = \frac{4}{12} = \frac{1}{3}$

The volume of a cone is given by the formula

$V = \frac{1}{3} \pi r^{2} h$

Substitute $r = \frac{1}{3}, h = \frac{3}{2}$ :

$V = \frac{1}{3}\cdot \left ( \frac{1}{3} \right ) ^{2}\cdot \frac{3}{2} \cdot \pi$

$V = \frac{1}{3}\cdot\frac{1}{3}\cdot \frac{1}{3}\cdot \frac{3}{2} \cdot \pi$

$V = \frac{1}{1}\cdot\frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{2} \cdot \pi$

$V = \frac{1}{18} \pi$

Give the volume of a cone whose height is 10 inches and whose base is a circle with circumference $6 \pi$ inches.

$30 \pi \textrm{ in}^{3}$

$90 \pi \textrm{ in}^{3}$

$45 \pi \textrm{ in}^{3}$

$360 \pi \textrm{ in}^{3}$

$120 \pi \textrm{ in}^{3}$

Explanation

A circle with circumference $6 \pi$ inches has as its radius

$r = \frac{C }{2\pi }=\frac{6\pi }{2\pi } = 3$ inches.

The area of the base is therefore

$B = \pi r^{2} = \pi \cdot 3^{2} = 9 \pi$ square inches.

To find the volume of the cone, substitute $B = 9 \pi , h = 10$ in the formula for the volume of a cone:

$V = \frac{1}{3} Bh = \frac{1}{3} \cdot 9 \pi \cdot 10 = 30 \pi$ cubic inches

Find the volume of a cone with the following measurements:

diameter = 12in
height = 6in

$72\pi \text{ in}^3$

$48\pi \text{ in}^3$

$36\pi \text{ in}^3$

$84\pi \text{ in}^3$

$54\pi \text{ in}^3$

Explanation

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

$V = \pi r^2 \frac{h}{3}$

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

Now, we know the diameter of the cone is 12in. We also know the diameter is two times the radius. Therefore, the radius is 6in.

We know the height of the cone is 6in.

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

$V = \pi \cdot (6\text{in})^2 \cdot \frac{6\text{in}}{3}$

$V = \pi \cdot 36\text{in}^2 \cdot 2\text{in}$

$V = \pi \cdot 72\text{in}^3$

$V = 72\pi \text{ in}^3$

You are an architect designing a cone shaped structure. If the structure will be 30 ft tall and 10 feet wide at the base, what will the volume of the structure be?

$250 \pi ft^3$

$100 \pi ft^3$

$22.5 \pi ft^3$

Explanation

You are an architect designing a cone shaped structure. If the structure will be 30 ft tall and 10 feet wide at the base, what will the volume of the structure be?

Begin by using the formula for volume of a cone:

$V=\pi r^2\frac{h}{3}$

Now, we simply need to plug in our knowns.

We know the height is 30 ft.

We know that the diameter is 10ft, however, we need the radius.

Divide 10 by 2 to get a radius of 5 ft.

Now, let's go....

$V=\pi (5ft)^2\frac{30ft}{3}=\pi25ft^2*10ft=250 \pi ft^3$

Find the volume of a cone with the following measurements:

height: 12in
diameter: 6in

$36\pi \text{ in}^3$

$24\pi \text{ in}^3$

$48\pi \text{ in}^3$

$32\pi \text{ in}^3$

$28\pi \text{ in}^3$

Explanation

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

$V = \pi r^2 \frac{h}{3}$

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

Now, we know the height of the cone is 12in. We also know the diameter of the cone is 6in. We know the diameter is two times the radius. Therefore, the radius is 3in.

So, we get

$V = \pi \cdot (3\text{in})^2 \cdot \frac{12\text{in}}{3}$