Math - How to find the volume of a cone

What is the equation of a circle with a center of (5,15) and a radius of 7?

Explanation

To find the equation of a circle we must first know the standard form of the equation of a circle which is

The letters and represent the -value and -value of the center of the circle respectively.

In this case is 5 and k is 15 so plugging the values into the equation yields

We then plug the radius into the equation to get

Square it to yield

The equation with a center of (5,15) and a radius of 7 is .

Find the volume of the following half cone.

Half_cone

$160 \pi m^3$

$150 \pi m^3$

$140 \pi m^3$

$170 \pi m^3$

$180 \pi m^3$

Explanation

The formula of the volume of a half cone is:

$V=\frac{1}{3}(base)(height)$

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

Where is the radius of the cone and is the height of the cone.

Use the Pythagorean Theorem to find the height of the cone:

$V = \frac{1}{3}\left(\frac{1}{2}\pi (8m)^2\right)(15m)$

$V=160 \pi m^3$

Find the volume of the following cone.

Cone

$96 \pi m^3$

$86 \pi m^3$

$76 \pi m^3$

$106 \pi m^3$

$116 \pi m^3$

Explanation

The formula for the volume of a cone is:

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

Where is the radius of the cone and is the height of the cone

Use the Pythagorean Theorem to find the length of the radius:

Plugging in our values, we get:

$V = \frac{\pi (6m)^2 8m}{3}$

$V = 96 \pi m^3$

What is the volume of a right cone with a diameter of 6 cm and a height of 5 cm?

$15\pi \ cm^{3}$

$120\pi \ cm^{3}$

$180\pi \ cm^{3}$

$45\pi \ cm^{3}$

$60\pi \ cm^{3}$

Explanation

The general formula is given by $V = 1/3Bh = 1/3\pi r^{2}h$ , where = radius and = height.

The diameter is 6 cm, so the radius is 3 cm.

$V=\frac{\pi 3^2\times 5}{3}=15\pi$

Find the volume of the following cone.

Cone

$100 \pi m^3$

$115 \pi m^3$

$130 \pi m^3$

$85 \pi m^3$

$70 \pi m^3$

Explanation

The formula for the volume of a cone is:

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

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

In order to find the height of the cone, use the Pythagorean Theorem:

Plugging in our values, we get:

$V= \frac{\pi (5m)^2 (12m)}{3}$

$V= \pi (25m^2)(4m)=100 \pi m^3$

$\textup{Find the volume, to the nearest cubic inch, of a cone with a radius of 8 inches}$

$\textup{at its base and a height of one foot.}$

Explanation

$V= \frac{1}{3} \pi r^{2}h;;\textup{Given: }r=8\textup{ inches}, h=12\textup{ inches}$

$V = \frac{1}{3} \pi\times8^{2}\times12;;;;;V = 256\pi;;;V=804.2$

There is a large cone with a radius of 4 meters and height of 18 meters. You can fill the cone with water at a rate of 3 cubic meters every 25 seconds. How long will it take you to fill the cone?

$41\ \textup{minutes}\ 53\ \textup{seconds}$

$3\ \textup{minutes}\ 20\ \textup{seconds}$

$800\ \textup{minutes}$

$600\ \textup{minutes}$

$60\ \textup{minutes}$

Explanation

First we will calculate the volume of the cone

$V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\cdot 4^{2}\cdot 18=96\pi\ m^{3}$

Next we will determine the time it will take to fill that volume

$96\pi \ m^{3}\times \frac{25\ s}{3\ m^{3}}=800\pi \ s$

We will then convert that into minutes

$800\pi \ s\times \frac{1\ minute}{60\ seconds}=13.\overline{3}\pi \ minutes\approx 41.9\ minutes\approx 41\ minutes\ 53\ seconds$

What is the volume of a cone with base radius 4, and height 6?

$\frac{96\pi}{3}$

$96\pi$

$\frac{16\pi}{3}$

$\frac{24}{3}$

Explanation

The volume of a cone is $\frac{\pi*r^2*h}{3}$ , where is the height of the cone and is the base radius.

The volume of this cone is thus:

$\frac{\pi*4^2*6}{3}$

= $\frac{96\pi}{3}$

What is the volume of a cone with a height of and a base with a radius of ?

$507\pi$

$117\pi$

$1521\pi$

$396\pi$

Explanation

To find the volume of a cone we must use the equation $V=\frac{1}{3}(A_{base})(H)$ . In this formula, $A_{base}$ is the area of the circular base of the cone, and is the height of the cone.