SOLUTION: An open rectangular tank (with no top) is to have a square base and a volume of 100 cubic feet. The cost per square foot for the bottom is $16, and for the sides is $10. What are t

Algebra ->  Finance -> SOLUTION: An open rectangular tank (with no top) is to have a square base and a volume of 100 cubic feet. The cost per square foot for the bottom is $16, and for the sides is $10. What are t      Log On


   

Question 1078723: An open rectangular tank (with no top) is to have a square base and a volume of 100 cubic feet. The cost per square foot for the bottom is $16, and for the sides is $10. What are the dimensions of the cheapest tank?
To achieve a minimum cost of $____________ , the tank should have a base of__________ feet by ____________feet and a height of ___________feet.
Please help me fill in the blanks as well. I would totally appreciate it.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
An open rectangular tank (with no top) is to have a square base and a volume of 100 cubic feet.
The cost per square foot for the bottom is $16, and for the sides is $10.
What are the dimensions of the cheapest tank?
:
Let x = the side of the square base
let h = the height of the tank
then
x^2 * h = 100 cu/ft
h = 100%2Fx%5E2
:
Surface area with no top
S.A. = x^2 + 4(h*x)
S.A. = x^2 + 4hx
Surface area cost
Total cost
C = 16x^2 + 10(4hx)
C = 16x^2 + 40hx
replace h with 100%2Fx%5E2
C = 16x^2 + 40(100%2Fx%5E2*x
cancel x
C = 16x^2 + 4000%2Fx
Plot a graph of this equation, total cost is on the y axis
+graph%28+300%2C+200%2C+-6%2C+20%2C+-1000%2C+5000%2C+16x%5E2%2B%284000%2Fx%29%2C+1200%29+
You can see that minimum cost occurs when x = 5 ft
Find the height
h = 100%2F5%5E2
h = 4 ft is the height for minimum cost
Therefor the dimensions of the tank 5 by 5 by 4
Find the actual cost
C = 16(5^2) + 10*4(5*4)
C = 400 + 800
C = $1200, the green line on the graph