How do you find the dimensions of the box that minimize the total cost of materials used if a rectangular milk carton box of width w, length l, and height h holds 534 cubic cm of milk and the sides of the box cost 4 cents per square cm and the top and bottom cost 8 cents per square cm?

Nathanael Perkins

Nathanael Perkins

Answered question

2022-09-27

How do you find the dimensions of the box that minimize the total cost of materials used if a rectangular milk carton box of width w, length l, and height h holds 534 cubic cm of milk and the sides of the box cost 4 cents per square cm and the top and bottom cost 8 cents per square cm?

Answer & Explanation

baselulaox

baselulaox

Beginner2022-09-28Added 8 answers

Note that varying the length and width to be other than equal blackuces 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:
Volume = w × l × h = w 2 h = 534
h = 534 w 2
Cost = (Cost of sides) + (Cost of top and bottom)
C = ( 4 × ( 4 w × h ) ) + ( 8 × ( 2 w 2 ) )
= ( 4 × ( 4 w × 534 w 2 ) + ( 8 × ( 2 w 2 ) )
= 8544 w - 1 + 16 w 2
d C d w = 0 for critical points
- 85434 w - 2 + 32 w = 0
Assuming w 0 we can multiply by w 2 and with some simple numeric division:
- 267 + w 3 = 0
and
w = ( 267 ) 1 3
=6.44 (approx.)
l = 6.44
and h=12.88 (approx.)

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Multivariable calculus

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?