Question
Michael has two different shaped containers. The dimensions of each container are shown in the drawing.
Cone is 2.5 in. At the top with a dot and the lines are going to the right. And there is another number 9 in.
The cylinder has 3 in. At the top and there is another number on the side it is an 8 in.
Which statement about the volumes of the cone and the cylinder is true?
Responses
A The volume of the cylinder is 167.29 cubic inches less than the volume of the cone.The volume of the cylinder is 167.29 cubic inches less than the volume of the cone.
B The volume of the cylinder is 48.89 cubic inches less than the volume of the cone. The volume of the cylinder is 48.89 cubic inches less than the volume of the cone.
C The volume of the cone is 48.89 cubic inches greater than the volume of the cylinder. The volume of the cone is 48.89 cubic inches greater than the volume of the cylinder.
D The volume of the cone is 167.29 cubic inches less than the volume of the cylinder.
Cone is 2.5 in. At the top with a dot and the lines are going to the right. And there is another number 9 in.
The cylinder has 3 in. At the top and there is another number on the side it is an 8 in.
Which statement about the volumes of the cone and the cylinder is true?
Responses
A The volume of the cylinder is 167.29 cubic inches less than the volume of the cone.The volume of the cylinder is 167.29 cubic inches less than the volume of the cone.
B The volume of the cylinder is 48.89 cubic inches less than the volume of the cone. The volume of the cylinder is 48.89 cubic inches less than the volume of the cone.
C The volume of the cone is 48.89 cubic inches greater than the volume of the cylinder. The volume of the cone is 48.89 cubic inches greater than the volume of the cylinder.
D The volume of the cone is 167.29 cubic inches less than the volume of the cylinder.
Answers
Answered by
GPT-4o mini
To solve the problem, we will calculate the volumes of both the cone and the cylinder using their respective formulas.
1. **Volume of the cone:**
The formula for the volume \( V \) of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
From the problem, the cone has a height (h) of 9 inches and a radius (r) of 2.5 inches.
- Thus, the volume of the cone is:
\[
V_{\text{cone}} = \frac{1}{3} \pi (2.5)^2 (9)
\]
\[
= \frac{1}{3} \pi (6.25)(9)
\]
\[
= \frac{1}{3} \pi (56.25)
\]
\[
= 18.75 \pi \approx 58.90 \text{ cubic inches}
\]
2. **Volume of the cylinder:**
The formula for the volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
From the problem, the cylinder has a height (h) of 8 inches and a radius (r) of 3 inches.
- Thus, the volume of the cylinder is:
\[
V_{\text{cylinder}} = \pi (3)^2 (8)
\]
\[
= \pi (9)(8)
\]
\[
= 72 \pi \approx 226.19 \text{ cubic inches}
\]
3. **Comparing the volumes:**
Now we can compare \( V_{\text{cone}} \) and \( V_{\text{cylinder}} \):
\[
V_{\text{cylinder}} - V_{\text{cone}} = 72\pi - 18.75\pi = (72 - 18.75)\pi \approx 53.25\pi \approx 167.24 \text{ cubic inches}
\]
Thus, we find:
- The volume of the cone is approximately 58.90 cubic inches.
- The volume of the cylinder is approximately 226.19 cubic inches.
- The volume of the cone is less than the volume of the cylinder by approximately 167.29 cubic inches.
The correct statement is **D:** The volume of the cone is 167.29 cubic inches less than the volume of the cylinder.
1. **Volume of the cone:**
The formula for the volume \( V \) of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
From the problem, the cone has a height (h) of 9 inches and a radius (r) of 2.5 inches.
- Thus, the volume of the cone is:
\[
V_{\text{cone}} = \frac{1}{3} \pi (2.5)^2 (9)
\]
\[
= \frac{1}{3} \pi (6.25)(9)
\]
\[
= \frac{1}{3} \pi (56.25)
\]
\[
= 18.75 \pi \approx 58.90 \text{ cubic inches}
\]
2. **Volume of the cylinder:**
The formula for the volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
From the problem, the cylinder has a height (h) of 8 inches and a radius (r) of 3 inches.
- Thus, the volume of the cylinder is:
\[
V_{\text{cylinder}} = \pi (3)^2 (8)
\]
\[
= \pi (9)(8)
\]
\[
= 72 \pi \approx 226.19 \text{ cubic inches}
\]
3. **Comparing the volumes:**
Now we can compare \( V_{\text{cone}} \) and \( V_{\text{cylinder}} \):
\[
V_{\text{cylinder}} - V_{\text{cone}} = 72\pi - 18.75\pi = (72 - 18.75)\pi \approx 53.25\pi \approx 167.24 \text{ cubic inches}
\]
Thus, we find:
- The volume of the cone is approximately 58.90 cubic inches.
- The volume of the cylinder is approximately 226.19 cubic inches.
- The volume of the cone is less than the volume of the cylinder by approximately 167.29 cubic inches.
The correct statement is **D:** The volume of the cone is 167.29 cubic inches less than the volume of the cylinder.