How to find the volume of a cone - Math

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Question

The radius of the base of a given cone is three times that of each base of a given cylinder. The heights of the cone and the cylinder are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

Answer

If we let be the radius of each base of the cylinder, then is the radius of the base of the cone. We can let be their common height.

In the formula for the volume of a cylinder, set and :

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

$V = \pi R ^{2}H$

In the formula for the volume of a cone, set and :

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

$V = $\frac{1}{3}$ \pi $(3R)^{2}$H = $\frac{1}{3}$ \pi $(9R^{2}$)H = 3\pi R ^{2}H$

, so $3\pi R ^{2}H > \pi R ^{2}H$ . The cone has the greater volume.

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Question

The height of Cone B is three times that of Cone A. The radius of the base of Cone B is one-half the radius of the base of Cone A.

Which is the greater quantity?

(a) The volume of Cone A

(b) The volume of Cone B

Answer

Let be the radius and height of Cone A, respectively. Then the radius and height of Cone B are $$\frac{1}{2}$r$ and , respectively.

(a) The volume of Cone A is $$\frac{1}{3}$ \pi $r^{2}$ h$ .

(b) The volume of Cone B is

$$\frac{1}{3}$ \pi \left ( $\frac{1}{2}$ r\right ) ^{2} \cdot 3 h = $\frac{1}{3}$ \cdot $\frac{1}{4}$ \cdot 3\cdot \pi r ^{2} h= $\frac{1}{4}$ r ^{2} h$ .

Since $$\frac{1}{4}$ r ^{2} h < $\frac{1}{3}$ r ^{2} h$ , the cone in (a) has the greater volume.

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Question

The volume of a cone whose height is three times the radius of its base is one cubic yard. Give its radius in inches.

Answer

The volume of a cone with base radius and height is

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

The height is three times this, or . Therefore, the formula becomes

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

$V = \pi $r^{3}$$

Set this volume equal to one and solve for :

$\pi $r^{3}$ = 1$

$\pi $r^{3}$ \div \pi = 1 \div \pi$

$r=\sqrt[3]{$\frac{ 1 }{ \pi}$} ={$\frac{ \sqrt[3] {1}$ }{ \sqrt[3]{ \pi}}} ={$\frac{ 1}{ \sqrt[3]{ \pi}$}}={$\frac{1 \cdot \sqrt[3]{ $\pi^{2}$$ }}{ \sqrt[3]{ \pi} \cdot \sqrt[3]{ $\pi^{2}$}} ={$\frac{ \sqrt[3]{ $\pi^{2}$$ }}{ \pi}}$

This is the radius in yards; since the radius in inches is requested, multiply by 36.

$$\frac{ \sqrt[3]{ $\pi^{2}$$ }}{ \pi} \times 36 = $\frac{ 36 \sqrt[3]{ $\pi^{2}$$ }}{ \pi}$

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Question

The height of a given cylinder is one half the height of a given cone. The radii of their bases are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

Answer

Call the radius of the base of the cone and the height of the cone. The cylinder will have bases of radius and height $$\frac{1}{2}$ H$ .

In the formula for the volume of a cylinder, set and $h = $\frac{1}{2}$ H$ :

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

$V = \pi R ^{2} \cdot $\frac{1}{2}$H = $\frac{1}{2}$ \pi R ^{2}H$

In the formula for the volume of a cone, set and :

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

$V = $\frac{1}{3}$ \pi $R^{2}$H$

$\frac{1}{2}$ > $\frac{1}{3}$ , so

$$\frac{1}{2}$ \pi R ^{2}H > $\frac{1}{3}$ \pi R ^{2}H$ ,

meaning that the cylinder has the greater volume.

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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 =

Cone Volume =

Cylinder Diameter = 6, therefore Cylinder Radius = 3

Cone Radius = 3

Empty Volume = Cylinder Volume – Cone Volume

$=\pi90-\pi30$

$=\pi*60$

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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 = 1/3pi $r^{2}$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 . In this formula, 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 . Using this, we can adjust our formula and plug in the value of our radius.

$V=\frac{1}{3}$(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 , where is the height of the cone and is the base radius.

The volume of this cone is thus:

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

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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

The radius of the base of a given cone is three times that of each base of a given cylinder. The heights of the cone and the cylinder are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

Answer

If we let be the radius of each base of the cylinder, then is the radius of the base of the cone. We can let be their common height.

In the formula for the volume of a cylinder, set and :

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

$V = \pi R ^{2}H$

In the formula for the volume of a cone, set and :

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

$V = $\frac{1}{3}$ \pi $(3R)^{2}$H = $\frac{1}{3}$ \pi $(9R^{2}$)H = 3\pi R ^{2}H$

, so $3\pi R ^{2}H > \pi R ^{2}H$ . The cone has the greater volume.

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Question

The height of Cone B is three times that of Cone A. The radius of the base of Cone B is one-half the radius of the base of Cone A.

Which is the greater quantity?

(a) The volume of Cone A

(b) The volume of Cone B

Answer

Let be the radius and height of Cone A, respectively. Then the radius and height of Cone B are $$\frac{1}{2}$r$ and , respectively.

(a) The volume of Cone A is $$\frac{1}{3}$ \pi $r^{2}$ h$ .

(b) The volume of Cone B is

$$\frac{1}{3}$ \pi \left ( $\frac{1}{2}$ r\right ) ^{2} \cdot 3 h = $\frac{1}{3}$ \cdot $\frac{1}{4}$ \cdot 3\cdot \pi r ^{2} h= $\frac{1}{4}$ r ^{2} h$ .

Since $$\frac{1}{4}$ r ^{2} h < $\frac{1}{3}$ r ^{2} h$ , the cone in (a) has the greater volume.

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Question

The volume of a cone whose height is three times the radius of its base is one cubic yard. Give its radius in inches.

Answer

The volume of a cone with base radius and height is

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

The height is three times this, or . Therefore, the formula becomes

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

$V = \pi $r^{3}$$

Set this volume equal to one and solve for :

$\pi $r^{3}$ = 1$

$\pi $r^{3}$ \div \pi = 1 \div \pi$

$r=\sqrt[3]{$\frac{ 1 }{ \pi}$} ={$\frac{ \sqrt[3] {1}$ }{ \sqrt[3]{ \pi}}} ={$\frac{ 1}{ \sqrt[3]{ \pi}$}}={$\frac{1 \cdot \sqrt[3]{ $\pi^{2}$$ }}{ \sqrt[3]{ \pi} \cdot \sqrt[3]{ $\pi^{2}$}} ={$\frac{ \sqrt[3]{ $\pi^{2}$$ }}{ \pi}}$

This is the radius in yards; since the radius in inches is requested, multiply by 36.

$$\frac{ \sqrt[3]{ $\pi^{2}$$ }}{ \pi} \times 36 = $\frac{ 36 \sqrt[3]{ $\pi^{2}$$ }}{ \pi}$

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Question

The height of a given cylinder is one half the height of a given cone. The radii of their bases are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

Answer

Call the radius of the base of the cone and the height of the cone. The cylinder will have bases of radius and height $$\frac{1}{2}$ H$ .

In the formula for the volume of a cylinder, set and $h = $\frac{1}{2}$ H$ :

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

$V = \pi R ^{2} \cdot $\frac{1}{2}$H = $\frac{1}{2}$ \pi R ^{2}H$

In the formula for the volume of a cone, set and :

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

$V = $\frac{1}{3}$ \pi $R^{2}$H$

$\frac{1}{2}$ > $\frac{1}{3}$ , so

$$\frac{1}{2}$ \pi R ^{2}H > $\frac{1}{3}$ \pi R ^{2}H$ ,

meaning that the cylinder has the greater volume.

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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 =

Cone Volume =

Cylinder Diameter = 6, therefore Cylinder Radius = 3

Cone Radius = 3

Empty Volume = Cylinder Volume – Cone Volume

$=\pi90-\pi30$

$=\pi*60$

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