of a right rectangular prism topped with a rectangular pyramid. The dimensions of the prism are 38 ft by 38 ft by 7 ft, A wedding tent is built in the shape of a right rectangular prism topped with a rectangular pyramid. The dimensions of the prism are 38 ft by 38 ft by 7 ft, and the height of the pyramid is 2 ft. Find the total volume of the tent. Round your answer to the nearest tenth if necessary.
The Correct Answer and Explanation is :
To find the total volume of the wedding tent, we need to calculate the volume of both the right rectangular prism and the rectangular pyramid on top, then sum them together.
Step 1: Calculate the Volume of the Rectangular Prism
The formula for the volume of a rectangular prism is:
[
V = \text{length} \times \text{width} \times \text{height}
]
Given:
- Length = 38 ft
- Width = 38 ft
- Height = 7 ft
[
V_{\text{prism}} = 38 \times 38 \times 7
]
[
V_{\text{prism}} = 10,108 \text{ ft}^3
]
Step 2: Calculate the Volume of the Rectangular Pyramid
The volume of a rectangular pyramid is given by:
[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
]
The base of the pyramid is the same as the top of the prism, so:
[
\text{Base Area} = 38 \times 38 = 1,444 \text{ ft}^2
]
Height of the pyramid = 2 ft.
[
V_{\text{pyramid}} = \frac{1}{3} \times 1,444 \times 2
]
[
V_{\text{pyramid}} = \frac{2,888}{3}
]
[
V_{\text{pyramid}} \approx 962.7 \text{ ft}^3
]
Step 3: Find the Total Volume
[
V_{\text{total}} = V_{\text{prism}} + V_{\text{pyramid}}
]
[
V_{\text{total}} = 10,108 + 962.7
]
[
V_{\text{total}} \approx 11,070.7 \text{ ft}^3
]
Final Answer:
The total volume of the wedding tent is 11,070.7 cubic feet.
Explanation (300 Words)
To determine the total volume of the wedding tent, we calculate the volume of two different geometric shapes: a right rectangular prism and a rectangular pyramid. The rectangular prism forms the main body of the tent, while the pyramid serves as the peaked roof.
First, we find the volume of the rectangular prism using the standard formula ( V = l \times w \times h ). Given that the length and width are both 38 ft and the height is 7 ft, the calculation results in a volume of 10,108 cubic feet.
Next, we calculate the volume of the rectangular pyramid using the formula ( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} ). The base area is determined by multiplying the length and width of the prism (since the base of the pyramid matches the top of the prism), which gives 1,444 square feet. Multiplying by the pyramid’s height of 2 ft, then dividing by 3, gives a pyramid volume of approximately 962.7 cubic feet.
Finally, adding both volumes together, we find the total volume of the wedding tent to be 11,070.7 cubic feet. This total volume represents the complete enclosed space within the tent, which is useful for determining factors such as air circulation, heating, or event capacity.