Problem 5 Suppose you want to construct an aquarium for your

Problem 5. Suppose you want to construct an aquarium for your pet shark and octopus. (Because, why not... However, your shark and octopus are not friends and as a result you need an aquarium that has a divider in the middle. The bottom of the aquarium is made out of slate that costs $10 dollars per square foot and the sides and divider are made out of glass which costs $5 dollars per square foot. The total volume must be 21,168 ft3 and each half tank is to be the same size (both with a square base) No To? At a. Construct a function C that gives the cost of the aquarium (the function should only depend on one variable). b. Minimize the function C constructed in part a. c. Find the dimensions that will minimize the cost of the aquarium

Solution

Volume, V =length * width *height =2x*x*y= 2x²y
21168=2x²y

y= 10584/x²

S = 2x*x+2 *y*x+2*2x*y = 2x²+ 6xy

a .Cost , C = 2x²*5 +6*5*xy = 10x²+30xy

b. Substituting the value of y

C = 10x²+30x(10584/x²) = 10x²+ (317520/x)

C \' = 20x -(317520/x²)

C \' =0

20x -(317520/x²)=0

x³=15876

x = 25.1 ft

y= 10834/x²= 10834/25.1²=17.2 ft