Optimization Problems

Question

A showroom for a car dealership is to be built in the shape of a rectangle with brick on the back and sides, and glass on the front. The floor of the showrrom is to have an area of 500 m².  

a)If a brick wall costs $1200/m while a glass wall costs $600/m, what dimensions would minimize the cost of the showroom?

Answer 1

Let dimensions of the floor be l and d. Then we know that ld=500; Cost of the wall is given by:

 

C=1200*(2d+l)+600*l=2400d+1800l; Using ld=500 we can get l=500/d, plug it in and obtain:

 

C(d)=2400d+1800*500/d=2400d+900000/d;

 

C'(d)=2400-900000/d²=0 gives the condition for the maximum/minimum;

 

Solving for d we obtain:

 

2400=900000/d² ⇔ d=√(9000/24)=√375=5√15 m; l=500/(5√15)=100/√15=20/3*√15 m

 

This value gives the total cost of the showroom:

C(5√15)=$2400*5√15+$600*20/3*√15=$16000√15≈$61968