Problema Solution
A closed box with square base and vertical sides is to be built to house an ant colony. The bottom of the box and all four sides are to be made of material costing $1/ft2, and the top is to be constructed of glass costing $6/ft2. What are the dimensions of the box of greatest volume that can be constructed for $84?
Answer provided by our tutors
let
x = be the length of the side of the squared base
y = the height of the box
the area of the bottom and the 4 sides is: x^2 + 4xy
the area of the top is: x^2
the total cost of the box is:
1*(x^2 + 4xy) + 6x^2 = 84
7x^2 + 4xy = 84
y = (1/4x)(84 - 7x^2)
we need to find the box of the greatest volume:
V = x^2y
plug y = (1/4x)(84 - 7x^2) into the last formula
V = x^2*((1/4x)(84 - 7x^2)
V = (1/4)(84x - 7x^3)
we need to find the value of x for which V is maximum
to find the maximum take the first derivative of V
V' = (1/4)(84 - 21x^2)
set it equal to 0 and solve
(1/4)(84 - 21x^2) = 0
by solving we find:
x = 2 ft
click here to see the step by step solution of the equation:
y = (1/(4*2))(84 - 7*2^2)
y = 7 ft
click here to see the step by step calculation:
the dimensions of the box of greatest volume are: the square base has a side with length 2 ft and the height of the box is 7 ft.