To find the height of the square prism with the same base area and volume as the square pyramid, we can start by using the formulas for the volume of both shapes.
The volume \( V \) of a square pyramid is given by the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Given the height of the square pyramid \( h_p = 6 \) units and the volume \( V = 70 \) units\(^3\), we can rearrange the formula to find the base area \( B \):
\[ 70 = \frac{1}{3} \times B \times 6 \]
Multiplying both sides by 3 gives:
\[ 210 = B \times 6 \]
Now, we can solve for \( B \):
\[ B = \frac{210}{6} = 35 \text{ units}^2 \]
The volume of the square prism is given by the formula:
\[ V = \text{Base Area} \times \text{Height of Prism} \]
Let \( h_{prism} \) be the height of the square prism. The volume of the prism is equal to the volume of the pyramid (70 units\(^3\)), so we have:
\[ 70 = B \times h_{prism} \]
Substituting the base area \( B = 35 \):
\[ 70 = 35 \times h_{prism} \]
Now, solve for \( h_{prism} \):
\[ h_{prism} = \frac{70}{35} = 2 \text{ units} \]
Therefore, the height of the square prism is 2 units.
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