How to find the volume of a cone - Math

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Question

You have an empty cylinder with a base diameter of 6 and a height of 10 and you have a cone full of water with a base radius of 3 and a height of 10. If you empty the cone of water into the cylinder, how much volume is left empty in the cylinder?

Answer

Cylinder Volume = $\pir^{2}h$

Cone Volume = $\frac{1}{3}\pir^{2}h$

Cylinder Diameter = 6, therefore Cylinder Radius = 3

Cone Radius = 3

Empty Volume = Cylinder Volume – Cone Volume

$=\pir^{2}h-\frac{1}{3}\pir^{2}h$

$=\pi3^{2}10-\frac{1}{3}\pi3^{2}10$

$=\pi90-\pi30$

$=\pi60$

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Question

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

Answer

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.

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

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Question

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?

Answer

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

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

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Question

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

Answer

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.

We must first solve for the area of the base using .

The equation for the area of a circle is $A=\pi(r^{2})$ . Using this, we can adjust our formula and plug in the value of our radius.

$V=\frac{1}{3}(A_{base})(H)=\frac{1}{3}(\pi r^2)(H)=\frac{1}{3}(\pi (13)^2)(H)$

$V=\frac{1}{3}(169\pi)(H)$

Now we can plug in our given height, .

$V=\frac{1}{3}(169\pi)(9)$

Multiply everything out to solve for the volume.

$V=\frac{1}{3}(169\pi)(9)=(169\pi)(3)=507\pi$

The volume of the cone is $507\pi$ .

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Question

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

Answer

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 .

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Question

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

Answer

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

The volume of this cone is thus:

$\frac{\pi4^26}{3}$

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

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Question

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

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

Answer

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

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

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Question

What is the volume of a cone that has a radius of 3 and a height of 4?

Answer

The standard equation for the volume of a cone is

$v=\frac{1}{3}\pi (r)^2h$

where denotes the radius and denotes the height.

Plug in the given values for and to find the answer:

$v=\frac{1}{3}\pi (3)^2(4)=\frac{1}{3}\pi (9)(4)=12\pi$

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Question

Find the volume of the following cone.

Answer

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:

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

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

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Question

Find the volume of the following cone.

Answer

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:

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

$V = 96 \pi m^3$

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Question

Find the volume of the following half cone.

Answer

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:

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

$V=160 \pi m^3$

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Question

You have an empty cylinder with a base diameter of 6 and a height of 10 and you have a cone full of water with a base radius of 3 and a height of 10. If you empty the cone of water into the cylinder, how much volume is left empty in the cylinder?

Answer

Cylinder Volume = $\pir^{2}h$

Cone Volume = $\frac{1}{3}\pir^{2}h$

Cylinder Diameter = 6, therefore Cylinder Radius = 3

Cone Radius = 3

Empty Volume = Cylinder Volume – Cone Volume

$=\pir^{2}h-\frac{1}{3}\pir^{2}h$

$=\pi3^{2}10-\frac{1}{3}\pi3^{2}10$

$=\pi90-\pi30$

$=\pi60$

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Question

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

Answer

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.

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

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Question

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?

Answer

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

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

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Question

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

Answer

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.

We must first solve for the area of the base using .

The equation for the area of a circle is $A=\pi(r^{2})$ . Using this, we can adjust our formula and plug in the value of our radius.

$V=\frac{1}{3}(A_{base})(H)=\frac{1}{3}(\pi r^2)(H)=\frac{1}{3}(\pi (13)^2)(H)$

$V=\frac{1}{3}(169\pi)(H)$

Now we can plug in our given height, .

$V=\frac{1}{3}(169\pi)(9)$

Multiply everything out to solve for the volume.

$V=\frac{1}{3}(169\pi)(9)=(169\pi)(3)=507\pi$

The volume of the cone is $507\pi$ .

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Question

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

Answer

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 .

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Question

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

Answer

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

The volume of this cone is thus:

$\frac{\pi4^26}{3}$

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

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Question

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

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

Answer

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

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

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Question

What is the volume of a cone that has a radius of 3 and a height of 4?

Answer

The standard equation for the volume of a cone is

$v=\frac{1}{3}\pi (r)^2h$

where denotes the radius and denotes the height.

Plug in the given values for and to find the answer:

$v=\frac{1}{3}\pi (3)^2(4)=\frac{1}{3}\pi (9)(4)=12\pi$

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Question

Find the volume of the following cone.

Answer

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:

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

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

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