Answer to Question #319513 in Calculus for Myca

Question

a. for the top and bottom of the box costs P30 per square centimeter and the material for the sides cost P15 per square centimeter. Find the dimensions of the box so that the total cost of material is the least possible and all its dimensions do not exceed 20cm.

Answer 1

Note that varying the length and width to be other than equal reduces the volume for the same total (length + width); or, stated another way,

w=l

for any optimal configuration.

Using given information about the Volume, express the height ( h) as a function of the width ( w ).

Write an expression for the Cost in terms of only the width ( w ).

Take the derivative of the Cost with respect to width and set it to zero to determine critical point(s).

Details:

$V=wlh=w^2h$

$h=V/w^2$

Cost = (Cost of sides) + (Cost of top and bottom

$C=4\times15hl+2\times30w^2=60V/w+60w^2$

$dC/dw=0$

$-60V/w^2+120w=0$

$w\not=0$

-60V+120w³=0

$w=\sqrt[3]{0.5V}$

Let h=20, then V=20w²

$w=2.16w^{2/3}$

$\sqrt[3]{w}=2.16$

w=10