Problema Solution
A close box with a square base is to be made out of 600 cm^2 of metal. Find the dimensions of the box so that its volume is a maximum. Find he value of this maximum volume.
Answer provided by our tutors
let
a = the length of the side of the square base
h = the height of the box
A = 600 cm^2 the are of the box
A = 2a^2 + 4ah
2a^2 + 4ah = 600
h = (600 - 2a^2)/(4a)
h = (300 - a^2)/(2a)
V = a^2*h
V = a^2*((300 - a^2)/(2a))
V = 150a - (1/2)a^3
we need to find the maximum of the function: y = (1/2)x^3 - 150x
y' = (3/2)x^2 - 150
by solving (3/2)x^2 - 150 = 0 we find:
x = 10
click here to see the step by step solution of the equation:
a = 10 cm
h = (300 - a^2)/(2a)
h = (300 - 10^2)/(2*10)
h = 200/20
h = 10 cm
V = 10^2*10
V = 1000 cm^3
the dimensions of the box are: the box is a cube with a side that has a length of 10 cm.
the volume of the cube is 1,000 cm^3.