Problema Solution
A rectangular box with a volume of 320 cubics feet is to be constructed with a square bas and top. the cost per square foot for the bottom is $.15, for the top is $.10 and for the sides is$.024. what dimension will minimize the cost?
Answer provided by our tutors
let
x = the length of the side of the squared base and top
y = the height
V = 320 ft^3 the volume of the rectangular box
the volume of the rectangular box is V = Area of the base * Height thus
V = x^2y
x^2y = 320
y = 320/x^2
the total cost is:
C = 0.15*x^2 + 0.10x^2 + 4*0.024*xy
plug y = 320/x^2 into the last equation:
C = 0.15*x^2 + 0.10x^2 + 4*0.024*x(320/x^2)
C = 0.25x^2 + 30.72/x
click here to see the step by step solution:
we need to find the minimum of the function:
C = 0.25x^2 + 30.72/x
click here to see the graph of the function:
the function has minimum for some x > 0
we find the minimum using derivatives:
C' = 0.5x - 30.72/x^2
C' = 0
by solving 0.5x - 30.72/x^2 = 0 we find:
x = 3.95 ft
click here to see the step by step solution of the equation:
the cost will be minimized for x = 3.95 ft
y = 320/3.95^2
y = 80.97 ft
the dimensions of the box are: the length of the side of the square base (and the top) is 3.85 ft, the height of the box is 80.97 ft.