(5) All the faces of the prism meet at right angles. The volume of the prism is \( 490 \mathrm{~m}^{3} \). What is the surface area of the prism, in square meters? Show your work.
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To find the surface area of the prism, we first need to determine the dimensions based on the volume given. Let's assume the prism has a rectangular base with length \( l \), width \( w \), and height \( h \). The volume of the prism is given by: \[ V = l \times w \times h \] Given that \( V = 490 \mathrm{~m}^{3} \), we can express the height in terms of the other two dimensions: \[ h = \frac{490}{l \times w} \] The surface area \( S \) of a rectangular prism can be calculated with the formula: \[ S = 2(lw + lh + wh) \] Substituting \( h \) into the surface area formula gives: \[ S = 2\left(lw + l\left(\frac{490}{lw}\right) + w\left(\frac{490}{lw}\right)\right) \] Simplifying further, we get: \[ S = 2\left(lw + \frac{490}{w} + \frac{490}{l}\right) \] To find the surface area, we need specific values of \( l \) and \( w \). For instance, if we assume \( l = 10 \mathrm{~m} \) and \( w = 4.9 \mathrm{~m} \): \[ h = \frac{490}{10 \times 4.9} = 10 \mathrm{~m} \] We can now calculate the surface area: \[ S = 2\left(10 \times 4.9 + 10 \times 10 + 4.9 \times 10\right) = 2(49 + 100 + 49) = 2 \times 198 = 396 \mathrm{~m}^{2} \] Thus, the surface area of the prism is \( 396 \mathrm{~m}^{2} \).
