2. A shed will have square ends and a sloped roof. The distance from the top of the square to the top peak is half the width of the square. The rectangular volume is 640 cubic feet. (This does not include any space in the triangular region). The cost of siding for all rectangular sides is $.75 per square foot. Nicer siding will be used in the triangular region. It will cost $1.50 per square foot. Shingles for roof will cost $1 per square foot.
a. Clearly define any variables used
b. Give an equation for volume of rectangular area.
c. Give the equation for the Surface area.
d. Give the equation for cost of siding and roofing the shed.
e. Give the equation for the derivative of cost.
f. Find the value of the derivative when the width of the square is 10 ft. Explain the meaning in terms of the problem.
g. Find when the derivative is equal to zero. Explain the significance of this point.
h. Give the dimensions of the shed that will minimize the cost of the shed.
i. Give the minimum cost of siding and roofing.