The management of the UNICO department store has decided to enclose an 873 ft2 a

Question

The management of the UNICO department store has decided to enclose an 873 ft2 area outside the building for displaying potted plants and flowers. One side will be formed by the external wall of the store, two sides will be constructed of pine boards, and the fourth side will be made of galvanized steel fencing. If the pine board fencing costs $7/running foot and the steel fencing costs $4/running foot, determine the dimensions of the enclosure that can be erected at minimum cost. (Round your answers to one decimal place.) wood side ft steel side ft

Explanation / Answer

First, we must model this as a system of two equations. We know area is x*y where x is the wood side and y is the steel. so we have x*y=873 ===> y = 873/x. Keep this for later. Now we must model the cost. We know that if we build this wall, we will need 2*x feet of wood and we know that the price of each foot is 7 so we have the price of wood portion to be 14x. The price of the steel portion is 4 y because it is 4 dollars per foot and we only have one side. So we have C = 14x+4y. We can plug in the y =873/x now to get 14x+3492/x = C. To find the cheapest, we find the derivative to be 14-3492/x^2 which we set equal to 0 and solve for x. This should get us x = 15.7933078 of wood. Solving for y, you should get 55.276577318 feet of steel

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