Question

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the design of a building that has a square pyramid roof as a roof is shown. the cost of material for the outside of the building and for the roof ranges from $25 per square foot to $50 per square foot. the budget for this material is $500,000. the rectangular front of the building has a length twice as long as its height. the slant height of the roof is the same as the height of the rectangular front of the building.
what is the maximum possible length of the rectangular front of the building to the nearest foot?
the maximum possible length of the rectangular front of the building is □ feet.
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Explanation:

Step1: Let the height of the rectangular front be \( h \) feet. Then the length of the rectangular front is \( l = 2h \) feet, and the slant height of the roof \( l_s=h \) feet.

Let's assume the cost per square - foot is the minimum (\( C = 25\) dollars per square - foot) to get the maximum area. The budget is \( B = 500000\) dollars. Using the formula \( B = C\times A\), we can find the maximum area \( A=\frac{B}{C}\). Substituting the values, \( A=\frac{500000}{25}=20000\) square feet.

Step2: Calculate the surface area of the building and roof.

The surface area of the four rectangular sides of the lower part of the building: The two side - rectangles have dimensions \( h\times2h\) and the two front - and - back rectangles have dimensions \( h\times2h\). The surface area of the four rectangular sides \(S_{1}=4\times(2h\times h)=8h^{2}\).
The surface area of the four triangular sides of the roof: The area of a single triangular side of the square - pyramid roof is \(A_{\triangle}=\frac{1}{2}\times(2h)\times h\), and for four triangular sides \(S_{2}=4\times\frac{1}{2}\times(2h)\times h = 4h^{2}\).
The total surface area \(S = S_{1}+S_{2}=8h^{2}+4h^{2}=12h^{2}\).

Step3: Solve for \( h \).

Since \(S = 12h^{2}\) and \(S = 20000\), we have the equation \(12h^{2}=20000\). Then \(h^{2}=\frac{20000}{12}=\frac{5000}{3}\approx1666.67\). So \(h=\sqrt{\frac{5000}{3}}\approx40.82\).

Step4: Find the length \( l \).

Since \(l = 2h\), substituting \(h\approx40.82\), we get \(l = 2\times40.82 = 81.64\approx82\)

Answer:

82