The volume of a cone is 27 cm3

Question

. What is the volume of a cylinder that shares the same radius and height as the cone?

Answer 1

The volume \( V \) of a cone is given by the formula:

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

where \( r \) is the radius and \( h \) is the height of the cone.

Given that the volume of the cone is \( 27 , \text{cm}^3 \), we can set up the equation:

\[ \frac{1}{3} \pi r^2 h = 27 \]

To find the volume of a cylinder with the same radius and height, we use the formula for the volume of a cylinder:

\[ V = \pi r^2 h \]

Notice that the volume of the cylinder is three times that of the cone because:

\[ \pi r^2 h = 3 \left(\frac{1}{3} \pi r^2 h\right) = 3 \cdot 27 , \text{cm}^3 = 81 , \text{cm}^3 \]

Therefore, the volume of the cylinder that shares the same radius and height as the cone is:

\[ \boxed{81 , \text{cm}^3} \]

Answer 2

The volume of a sphere is 72 m3
. What is the volume of a cone if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter

Answer 3

To find the volume of the cone, we first need to determine the radius of the sphere from its volume.

The volume \( V \) of a sphere is given by the formula:

\[ V = \frac{4}{3} \pi r^3 \]

Given that the volume of the sphere is \( 72 , \text{m}^3 \), we can set up the equation:

\[ \frac{4}{3} \pi r^3 = 72 \]

To solve for \( r^3 \), we first multiply both sides by \( \frac{3}{4 \pi} \):

\[ r^3 = \frac{72 \cdot 3}{4 \pi} = \frac{216}{4 \pi} = \frac{54}{\pi} \]

Now, we can find the radius \( r \):

\[ r = \sqrt[3]{\frac{54}{\pi}} \]

Next, we need to determine the height of the cone. The diameter of the sphere is given by:

\[ \text{Diameter} = 2r \]

Now we can find the volume \( V \) of the cone using its formula:

The volume \( V \) of a cone is given by:

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

Substituting \( h = 2r \):

\[ V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

Now we can substitute \( r^3 \) from above:

\[ V = \frac{2}{3} \pi \left(\frac{54}{\pi}\right) \]

This simplifies as follows:

\[ V = \frac{2 \cdot 54}{3} = \frac{108}{3} = 36 , \text{m}^3 \]

Therefore, the volume of the cone is:

\[ \boxed{36 , \text{m}^3} \]

Answer 4

A 3D modeler is designing a component of a castle in a video game. She places a cone on top of a cylinder so they line up and share the same radius. The cone and the cylinder share the same height. If the volume of the cone is 50 cm3
, what is the volume of the whole object?

Answer 5

To find the volume of the whole object consisting of a cone on top of a cylinder, we first need to calculate the volume of the cylinder and then add it to the volume of the cone.

Step 1: Volume of the Cone

The volume \( V_c \) of the cone is given, which is:

\[ V_c = 50 , \text{cm}^3 \]

Step 2: Determine the Dimensions

Using the formula for the volume of a cone:

\[ V_c = \frac{1}{3} \pi r^2 h \]

we can express \( h \) in terms of \( r \):

\[ 50 = \frac{1}{3} \pi r^2 h \]

\[ h = \frac{150}{\pi r^2} \]

Step 3: Volume of the Cylinder

The volume \( V_cyl \) of the cylinder is given by:

\[ V_{cyl} = \pi r^2 h \]

We can substitute the expression for \( h \) derived from the volume of the cone into the cylinder's volume formula:

\[ V_{cyl} = \pi r^2 \left(\frac{150}{\pi r^2}\right) \]

Simplifying this, we have:

\[ V_{cyl} = 150 , \text{cm}^3 \]

Step 4: Total Volume of the Whole Object

Now we can find the total volume \( V_{total} \) of the whole object (cone + cylinder):

\[ V_{total} = V_c + V_{cyl} \]

\[ V_{total} = 50 , \text{cm}^3 + 150 , \text{cm}^3 = 200 , \text{cm}^3 \]

Final Result

The volume of the whole object is:

\[ \boxed{200 , \text{cm}^3} \]

Answer 6

Suppose a sphere, cylinder, and cone all share the same radius. The cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?(1 point)
Responses

sphere = cylinder + cone
sphere = cylinder + cone

cylinder = sphere – cone
cylinder = sphere – cone

cone = cylinder – sphere
cone = cylinder – sphere

cone = sphere – cylinder

Answer 7

Let's find the volumes of the sphere, cylinder, and cone based on the given conditions.

Volumes of the Shapes

1. Sphere: The volume \( V_s \) of a sphere is given by: \[ V_s = \frac{4}{3} \pi r^3 \]

2. Cylinder: The volume \( V_c \) of a cylinder (with height \( h = 2r \) and radius \( r \)) is given by: \[ V_c = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]

3. Cone: The volume \( V_{cone} \) of a cone (with height \( h = 2r \) and radius \( r \)) is given by: \[ V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

Relationship Between the Volumes

To compare the volumes, we can express them in relation to each other:

1. Total Volume (Cylinder + Cone): \[ V_c + V_{cone} = 2\pi r^3 + \frac{2}{3} \pi r^3 = \left(2 + \frac{2}{3}\right) \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

2. Volume of Sphere: \[ V_s = \frac{4}{3} \pi r^3 \]

Comparing Volumes

We see that:

• \( V_c + V_{cone} = \frac{8}{3} \pi r^3 \)
• \( V_s = \frac{4}{3} \pi r^3 \)

Thus we can express the relationship: \[ V_c + V_{cone} = 2V_s \]

Analyzing the Response Choices

Considering the options you provided:

1. sphere = cylinder + cone (False)
2. cylinder = sphere – cone (False)
3. cone = cylinder – sphere (False)
4. cone = sphere – cylinder (False)

None of these statements express the accurate relationship we derived. However, we found that:

\[ V_c + V_{cone} = 2 V_s \]

If that were an option, it would be correct, but given the available choices, none of them accurately reflect the relationships.

Thus, the best conclusion from the choices provided would be that none of the options are correct based on the derived relationships.

Answer 8

Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cm3
. What is the volume of a sphere if its radius is the same as the cylinder’s and the height of the cylinder is equal to the sphere’s diameter?

Answer 9

To find the volume of the sphere given that the volume of the cylinder is \(36 , \text{cm}^3\), we need to relate the dimensions of the cylinder and the sphere based on the problem's specifications.

Step 1: Cylinder Specifications

Let \( r \) be the radius of both the cylinder and the sphere. The height \( h \) of the cylinder is given as:

\[ h = \text{Diameter of the sphere} = 2r \]

Step 2: Volume of the Cylinder

The volume \( V_c \) of a cylinder is given by:

\[ V_c = \pi r^2 h \]

Substituting the height \( h = 2r \):

\[ V_c = \pi r^2 (2r) = 2\pi r^3 \]

Given that the volume of the cylinder is \( 36 , \text{cm}^3 \):

\[ 2\pi r^3 = 36 \]

Now we can solve for \( r^3 \):

\[ \pi r^3 = 18 \] \[ r^3 = \frac{18}{\pi} \]

Step 3: Volume of the Sphere

The volume \( V_s \) of a sphere is given by:

\[ V_s = \frac{4}{3} \pi r^3 \]

Substituting \( r^3 = \frac{18}{\pi} \):

\[ V_s = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) \]

This simplifies to:

\[ V_s = \frac{4}{3} \cdot 18 = \frac{72}{3} = 24 , \text{cm}^3 \]

Final Result

Thus, the volume of the sphere is:

\[ \boxed{24 , \text{cm}^3} \]