Problema Solution
Carla is a carpenter who has been hired to make a closed box with a square base and volume of 250 cubic meters. The material for the top and bottom of the box costs $2 per square meter, and the material for the sides cost $1 per square meter. Can Carla construct the box for less then $300?
Answer provided by our tutors
let
a = the side of the square, a>0
h = the heights of the box, h >0
V = (a^2)*h is the volume of the box and V = 250 m^3
(a^2)*h = 250
h = 250/(a^2)
we want the cost to be less then $300
2*2*(a^2) + 1*4*a*h < 300 divide both sides by 4
a^2 + a*h < 75
a^2 + a*250/(a^2) - 75 < 0 multiply both sides by a
a^3 - 75a + 250 < 0
a^3 - 75a + 250 = ((a - 5)^2)*(a + 10)
((a - 5)^2)*(a + 10) < 0
since (a - 5)^2 > 0 for every a we have
a + 10 < 0
a < - 10
we get contradiction since a > 0 thus Carla can not construct the box for less then $300.
you will get the same conclusion if you plot the graph of the function f(x) = x^3 - 75x + 250 and look for its extremes.
the function f(x) = x^3 - 75x + 25 has a local extreme for x = 5, f(5) = 0 for every x>0 BUT we need f(x) < 0 thus such box can not be constructed.