Question
A square prism with a pyramid on top has a total height of 25 feet. Each edge of the base is 8 feet
long. What is the total volume of the figure if the height of the pyramid is exactly one-fifth the total
height?
long. What is the total volume of the figure if the height of the pyramid is exactly one-fifth the total
height?
Answers
Answered by
GPT-4o mini
To find the total volume of the figure consisting of a square prism with a pyramid on top, we first need to calculate the volume of both the prism and the pyramid separately, and then sum these volumes.
### Step 1: Calculate the volume of the square prism.
The base of the prism is a square with each edge measuring 8 feet. Therefore, the area \( A \) of the base is:
\[
A = \text{side} \times \text{side} = 8 \, \text{ft} \times 8 \, \text{ft} = 64 \, \text{ft}^2.
\]
Next, we calculate the height of the square prism. The total height of the figure is 25 feet, and the height of the pyramid is one-fifth of the total height. Thus, we first find the height of the pyramid:
\[
\text{Height of the pyramid} = \frac{1}{5} \times 25 \, \text{ft} = 5 \, \text{ft}.
\]
Now we can determine the height of the prism:
\[
\text{Height of the prism} = \text{Total height} - \text{Height of the pyramid} = 25 \, \text{ft} - 5 \, \text{ft} = 20 \, \text{ft}.
\]
Now, we can calculate the volume \( V_p \) of the prism:
\[
V_p = \text{Base Area} \times \text{Height of the prism} = 64 \, \text{ft}^2 \times 20 \, \text{ft} = 1280 \, \text{ft}^3.
\]
### Step 2: Calculate the volume of the pyramid.
The volume \( V_y \) of the pyramid is given by the formula:
\[
V_y = \frac{1}{3} \times \text{Base Area} \times \text{Height of the pyramid}.
\]
The base area of the pyramid is the same as that of the prism, which is \( 64 \, \text{ft}^2 \), and the height of the pyramid is \( 5 \, \text{ft} \):
\[
V_y = \frac{1}{3} \times 64 \, \text{ft}^2 \times 5 \, \text{ft} = \frac{1}{3} \times 320 \, \text{ft}^3 = \frac{320}{3} \, \text{ft}^3 \approx 106.67 \, \text{ft}^3.
\]
### Step 3: Calculate the total volume.
Finally, we sum the volumes of the prism and the pyramid:
\[
V_{\text{total}} = V_p + V_y = 1280 \, \text{ft}^3 + \frac{320}{3} \, \text{ft}^3.
\]
To add these volumes together, convert \( 1280 \, \text{ft}^3 \) into a fraction:
\[
1280 = \frac{3840}{3} \, \text{ft}^3.
\]
Now add the volumes:
\[
V_{\text{total}} = \frac{3840}{3} \, \text{ft}^3 + \frac{320}{3} \, \text{ft}^3 = \frac{3840 + 320}{3} \, \text{ft}^3 = \frac{4160}{3} \, \text{ft}^3.
\]
Thus, the total volume of the figure is
\[
V_{\text{total}} = \frac{4160}{3} \, \text{ft}^3 \approx 1386.67 \, \text{ft}^3.
\]
### Final Answer:
The total volume of the figure is
\[
\frac{4160}{3} \, \text{ft}^3 \quad \text{or} \quad 1386.67 \, \text{ft}^3.
\]
### Step 1: Calculate the volume of the square prism.
The base of the prism is a square with each edge measuring 8 feet. Therefore, the area \( A \) of the base is:
\[
A = \text{side} \times \text{side} = 8 \, \text{ft} \times 8 \, \text{ft} = 64 \, \text{ft}^2.
\]
Next, we calculate the height of the square prism. The total height of the figure is 25 feet, and the height of the pyramid is one-fifth of the total height. Thus, we first find the height of the pyramid:
\[
\text{Height of the pyramid} = \frac{1}{5} \times 25 \, \text{ft} = 5 \, \text{ft}.
\]
Now we can determine the height of the prism:
\[
\text{Height of the prism} = \text{Total height} - \text{Height of the pyramid} = 25 \, \text{ft} - 5 \, \text{ft} = 20 \, \text{ft}.
\]
Now, we can calculate the volume \( V_p \) of the prism:
\[
V_p = \text{Base Area} \times \text{Height of the prism} = 64 \, \text{ft}^2 \times 20 \, \text{ft} = 1280 \, \text{ft}^3.
\]
### Step 2: Calculate the volume of the pyramid.
The volume \( V_y \) of the pyramid is given by the formula:
\[
V_y = \frac{1}{3} \times \text{Base Area} \times \text{Height of the pyramid}.
\]
The base area of the pyramid is the same as that of the prism, which is \( 64 \, \text{ft}^2 \), and the height of the pyramid is \( 5 \, \text{ft} \):
\[
V_y = \frac{1}{3} \times 64 \, \text{ft}^2 \times 5 \, \text{ft} = \frac{1}{3} \times 320 \, \text{ft}^3 = \frac{320}{3} \, \text{ft}^3 \approx 106.67 \, \text{ft}^3.
\]
### Step 3: Calculate the total volume.
Finally, we sum the volumes of the prism and the pyramid:
\[
V_{\text{total}} = V_p + V_y = 1280 \, \text{ft}^3 + \frac{320}{3} \, \text{ft}^3.
\]
To add these volumes together, convert \( 1280 \, \text{ft}^3 \) into a fraction:
\[
1280 = \frac{3840}{3} \, \text{ft}^3.
\]
Now add the volumes:
\[
V_{\text{total}} = \frac{3840}{3} \, \text{ft}^3 + \frac{320}{3} \, \text{ft}^3 = \frac{3840 + 320}{3} \, \text{ft}^3 = \frac{4160}{3} \, \text{ft}^3.
\]
Thus, the total volume of the figure is
\[
V_{\text{total}} = \frac{4160}{3} \, \text{ft}^3 \approx 1386.67 \, \text{ft}^3.
\]
### Final Answer:
The total volume of the figure is
\[
\frac{4160}{3} \, \text{ft}^3 \quad \text{or} \quad 1386.67 \, \text{ft}^3.
\]
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