Applied Calc - A hardware store owner chooses to enclose an 800 square foot rectangular area in front of her store so that one of the sides of the store will be used as one of the four sides of the fence. If the two sides that come out from the store front cost $3 per running foot for materials and the side parallel to the store front costs $5 per running foot for materials, then find the dimensions of the fence that will minimize the cost to construct the fence. Round all dimensions to the nearest foot.

Question

Applied Calc - A hardware store owner chooses to enclose an 800 square foot rectangular area in front of her store so that one of the sides of the store will be used as one of the four sides of the fence.  If the two sides that come out from the store front cost $3 per running foot for materials and the side parallel to the store front costs $5 per running foot for materials, then find the dimensions of the fence that will minimize the cost to construct the fence.  Round all dimensions to the nearest foot.

anonymous · Answer

ok, lots of info to take in on this one.  First we are given the area of the enclosure:
A=l*w=800
We know that the perimeter of a rectangle is:
P=2l+2w
Since one of the sides of the enclosure is the store, it doesn't add to the cost of the enclosure.  We can express the cost of the enclose as follows:
C=5l+3(2w)=5l+6w
Using our expression for area, we express this as a function of w alone
C(w)=(4000/w)+6w
We want to minimize this function.  Taking first derivative and setting equal to zero:
C'(w)=6-(4000/w^2)=0
0=6w^2-4000
w=25.8 ft
So the length is l=800/25.8=31.0 ft

anonymous · Answer

Thanks!

anonymous · Answer

np hope it's correct!

anonymous · Answer

It makes sense once you typed it out step by step.