Question
The figure is a cone with a sphere within it.
To the nearest whole number, what is the approximate volume of the shaded part of this figure?
Use 3.14 for Pi.
Drag the correct value to the box.
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Response area ft³
The circle of the triangle is 3ft and the bottom on the triangle is 6ft and the side length on the traignle and the bottom is 6ft
A. 14
B. 42
C. 57
D. 113
E. 156
To the nearest whole number, what is the approximate volume of the shaded part of this figure?
Use 3.14 for Pi.
Drag the correct value to the box.
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Response area ft³
The circle of the triangle is 3ft and the bottom on the triangle is 6ft and the side length on the traignle and the bottom is 6ft
A. 14
B. 42
C. 57
D. 113
E. 156
Answers
GPT-4o mini
To find the volume of the shaded part of the figure, we need to calculate the volume of the cone and then subtract the volume of the sphere inside it.
**Step 1: Calculate the volume of the cone.**
The formula for the volume \( V \) of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
Where:
- \( r \) is the radius of the base of the cone.
- \( h \) is the height of the cone.
From the given dimensions, the diameter of the base of the cone is 3 ft, so the radius \( r \) is:
\[
r = \frac{3}{2} = 1.5 \text{ ft}
\]
The height \( h \) of the cone is given as 6 ft.
Plugging in the values:
\[
V_{\text{cone}} = \frac{1}{3} \times 3.14 \times (1.5)^2 \times 6
\]
Calculating \( (1.5)^2 = 2.25 \):
\[
V_{\text{cone}} = \frac{1}{3} \times 3.14 \times 2.25 \times 6
\]
Now calculating:
\[
V_{\text{cone}} = \frac{1}{3} \times 3.14 \times 13.5
\]
\[
V_{\text{cone}} = \frac{1}{3} \times 42.39 \approx 14.13 \text{ ft}^3
\]
**Step 2: Calculate the volume of the sphere.**
The formula for the volume \( V \) of a sphere is:
\[
V = \frac{4}{3} \pi r^3
\]
The radius of the sphere is the same as the radius of the base of the cone, which is 1.5 ft.
Plugging in the values:
\[
V_{\text{sphere}} = \frac{4}{3} \times 3.14 \times (1.5)^3
\]
Calculating \( (1.5)^3 = 3.375 \):
\[
V_{\text{sphere}} = \frac{4}{3} \times 3.14 \times 3.375
\]
\[
V_{\text{sphere}} = \frac{4}{3} \times 10.615 \approx 14.15 \text{ ft}^3
\]
**Step 3: Calculate the volume of the shaded part.**
Volume of the shaded part = Volume of the cone - Volume of the sphere
\[
V_{\text{shaded}} = V_{\text{cone}} - V_{\text{sphere}}
\]
\[
V_{\text{shaded}} \approx 14.13 - 14.15 \approx -0.02 \text{ ft}^3
\]
Since the volume of the sphere seems to be increasing, the values may not give us a positive shaded volume. In practice, you want to find which option is closest.
Given your answer options:
A. 14
B. 42
C. 57
D. 113
E. 156
The closest (and best) estimate given your values would be **A. 14 ft³**, as this corresponds directly to the volume of the cone.
If you meant to calculate how much sphere volume is contained, re-examine the dimensions or clarify the situation, as this calculation leads to zero or negative shaded volume. However, **the apparent volume of the cone alone is the best candidate being 14 ft³.**
**Step 1: Calculate the volume of the cone.**
The formula for the volume \( V \) of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
Where:
- \( r \) is the radius of the base of the cone.
- \( h \) is the height of the cone.
From the given dimensions, the diameter of the base of the cone is 3 ft, so the radius \( r \) is:
\[
r = \frac{3}{2} = 1.5 \text{ ft}
\]
The height \( h \) of the cone is given as 6 ft.
Plugging in the values:
\[
V_{\text{cone}} = \frac{1}{3} \times 3.14 \times (1.5)^2 \times 6
\]
Calculating \( (1.5)^2 = 2.25 \):
\[
V_{\text{cone}} = \frac{1}{3} \times 3.14 \times 2.25 \times 6
\]
Now calculating:
\[
V_{\text{cone}} = \frac{1}{3} \times 3.14 \times 13.5
\]
\[
V_{\text{cone}} = \frac{1}{3} \times 42.39 \approx 14.13 \text{ ft}^3
\]
**Step 2: Calculate the volume of the sphere.**
The formula for the volume \( V \) of a sphere is:
\[
V = \frac{4}{3} \pi r^3
\]
The radius of the sphere is the same as the radius of the base of the cone, which is 1.5 ft.
Plugging in the values:
\[
V_{\text{sphere}} = \frac{4}{3} \times 3.14 \times (1.5)^3
\]
Calculating \( (1.5)^3 = 3.375 \):
\[
V_{\text{sphere}} = \frac{4}{3} \times 3.14 \times 3.375
\]
\[
V_{\text{sphere}} = \frac{4}{3} \times 10.615 \approx 14.15 \text{ ft}^3
\]
**Step 3: Calculate the volume of the shaded part.**
Volume of the shaded part = Volume of the cone - Volume of the sphere
\[
V_{\text{shaded}} = V_{\text{cone}} - V_{\text{sphere}}
\]
\[
V_{\text{shaded}} \approx 14.13 - 14.15 \approx -0.02 \text{ ft}^3
\]
Since the volume of the sphere seems to be increasing, the values may not give us a positive shaded volume. In practice, you want to find which option is closest.
Given your answer options:
A. 14
B. 42
C. 57
D. 113
E. 156
The closest (and best) estimate given your values would be **A. 14 ft³**, as this corresponds directly to the volume of the cone.
If you meant to calculate how much sphere volume is contained, re-examine the dimensions or clarify the situation, as this calculation leads to zero or negative shaded volume. However, **the apparent volume of the cone alone is the best candidate being 14 ft³.**