Arturo builds a storage box with a square base. The box should have a volume of 80 cubic centimeters.

Question

Arturo builds a storage box with a square base. The box should have a volume of 80 cubic centimeters. The material used for the top and bottom of the box (the square bases) cost Php 10.00 per square centimeter, while the lateral faces cost Php 15.00 per square centimeter. What dimensions will minimize the cost of the box?

Answer 1

V= hx^2 = 80 cm^3, h=80/x^2

A= 2B+4S (2 times area of the bottom plus 4 times the area of the sides)

A=2x^2 + 4xh = 2x^2 + 4x(80/x^2) = 2x^2 + 320/x

C=10(2x^2) + 15(320/x) = 20x^2 + 4800/x

C'(x) = 40x -4800/x^2 = 0

4800/x^2 = 40x

40x^3 = 4800

x^3 = 120

x = cube root of 120 =about 4.9 cm

h= about 80/24= 3.3 cm

4.9 by 4.9 by 3.3 = 79.2. so make the dimensions slighter larger than

4.9 by 4.9 by 3.3

C(4.9) = 15(2)(4.9)^2 + 10(4)(3.3)(4.9) = about 720 + 2134.5 = 2,854.5 Php

is Php Philippines pesos? Arturo sounds spanish, but then the Philippines are named after Spain's King Phillip. I think days of the week are the same in Spanish as in Tagalog.