To compare the volumes of the triangular pyramid and triangular prism, we can use the formulas for their volumes.
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Volume of the Pyramid: \[ V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
The base is a triangle, and we can calculate its area using the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Since we don't have the height of the triangle, we use Heron's formula instead.
First, we need the semi-perimeter (s) of the triangle: \[ s = \frac{3 + 5 + 6}{2} = 7 \] The sides are 3, 5, and we can find the third side using the Pythagorean Theorem in a 3-4-5 right triangle: \[ c = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = 3 \]
Now, we can calculate the area using Heron's formula: \[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{7(7-3)(7-5)(7-6)} = \sqrt{7 \times 4 \times 2 \times 1} = \sqrt{28} \approx 5.29 \] Thus, \[ V_{\text{pyramid}} = \frac{1}{3} \times 5.29 \times 8.41 \approx 14.85 , \text{inches}^3 \]
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Volume of the Prism: \[ V_{\text{prism}} = \text{Base Area} \times \text{Height} \] Using the base area calculated earlier, \[ V_{\text{prism}} = 5.29 \times 8.41 \approx 44.45 , \text{inches}^3 \]
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Comparing Volumes: To find out how much larger the volume of the prism is compared to the pyramid, \[ \text{Difference} = V_{\text{prism}} - V_{\text{pyramid}} \approx 44.45 - 14.85 \approx 29.60 , \text{inches}^3 \]
Thus, the prism has a larger volume than the pyramid, and it is approximately 3 times larger.
The correct response is:
- The prism has a larger volume by three times.