Asked by kymari davis-shanks
Volume of Cones, Cylinders, and Spheres Practice
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Question
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A 3-dimensional cone is placed on top of a 3-dimensional cylinder. One of the cylinder bases is common to the cone, with the same circumference.
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?
(1 point)
cm3
Complete this assessment to review what you've learned. It will not count toward your grade.
5 of 55 of 5 Items
Question
Use the image to answer the question.
A 3-dimensional cone is placed on top of a 3-dimensional cylinder. One of the cylinder bases is common to the cone, with the same circumference.
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?
(1 point)
cm3
Answers
Answered by
GPT-4o mini
To find the volume of the whole object made up of a cone and a cylinder, we need to calculate the volume of the cylinder and then add it to the volume of the cone.
The volume \( V \) of a cone is given by the formula:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
And the volume \( V \) of a cylinder is given by the formula:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
From the information given, we know:
- The volume of the cone \( V_{\text{cone}} = 50 \, \text{cm}^3 \)
- The height of the cone and the cylinder are equal, which we'll call \( h \).
- The radius of both the cone and the cylinder are the same, which we will call \( r \).
Since the cone's volume is already provided, we can express the height in terms of \( r \) using the formula for the volume of the cone:
\[
50 = \frac{1}{3} \pi r^2 h
\]
To find \( h \), we can rearrange this:
\[
h = \frac{150}{\pi r^2}
\]
Now we substitute this height into the formula for the cylinder's volume.
\[
V_{\text{cylinder}} = \pi r^2 h = \pi r^2 \left(\frac{150}{\pi r^2}\right) = 150 \, \text{cm}^3
\]
Now we can find the total volume of the whole object, which is the sum of the volume of the cone and the volume of the cylinder:
\[
V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}} = 50 + 150 = 200 \, \text{cm}^3
\]
Thus, the volume of the whole object is **200 cm³**.
The volume \( V \) of a cone is given by the formula:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
And the volume \( V \) of a cylinder is given by the formula:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
From the information given, we know:
- The volume of the cone \( V_{\text{cone}} = 50 \, \text{cm}^3 \)
- The height of the cone and the cylinder are equal, which we'll call \( h \).
- The radius of both the cone and the cylinder are the same, which we will call \( r \).
Since the cone's volume is already provided, we can express the height in terms of \( r \) using the formula for the volume of the cone:
\[
50 = \frac{1}{3} \pi r^2 h
\]
To find \( h \), we can rearrange this:
\[
h = \frac{150}{\pi r^2}
\]
Now we substitute this height into the formula for the cylinder's volume.
\[
V_{\text{cylinder}} = \pi r^2 h = \pi r^2 \left(\frac{150}{\pi r^2}\right) = 150 \, \text{cm}^3
\]
Now we can find the total volume of the whole object, which is the sum of the volume of the cone and the volume of the cylinder:
\[
V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}} = 50 + 150 = 200 \, \text{cm}^3
\]
Thus, the volume of the whole object is **200 cm³**.