Question

A landscape architect plans to enclose a square foot rectangular region in a botanical garden. She will use shrubs costing per foot along three sides and fencing costing per foot along the fourth side. Find the minimum total cost. Round the answer to the nearest cent.v

Answer 1

If the rectangular region has dimensions x and y, then its area is A = xy = 3000ft2.(let the area be 3000 sqft)

So y = 3000 x . If y is the side with fencing costing (let)$10 per foot, then the cost for this side is $ 10 y. The cost for the three other sides, where shrubs costing (let)$15 is used, is then $ 15 (2x+y).

Therefore the total cost is: C(x) = 10y + 15(2x + y) = 30x + 25y.

Since y = 3000/ x , then C(x) = 30x + 25* 3000/ x that we wish to minimize. Since C '(x) = 30 − 25 *3000/ x2 , then C'(x) = 0 for x 2 = 25* 3000/ 30 = 2500. Therefore, since x is positive, we have only one critical number in the domain which is x = 50ft . Since C''(x) = 25 *1500 /x3 , we have C ''(50) > 0. Thus, by the 2nd derivative test, C has a local minimum at x = 50, and therefore an absolute minimum because we have only one critical number in the domain. Hence, the minimum cost is C(50) = $4500, with the dimensions x = 50 ft and y = 3000 /50 = 60 ft.

They all are assumed value on the assumption as area to be 3000sqft,cost of side as $10 and cost of shrub as $15.