A box with a square base and open top must have a volume of 32,000cm^3. How do you find the dimensions of the box that minimize the amount of material used?

[ad_1]

The Volume of a box with a square base ##x## by ##x## cm and height ##h## cm is ##V=x^2h##

Don't use plagiarized sources. Get Your Custom Essay on
A box with a square base and open top must have a volume of 32,000cm^3. How do you find the dimensions of the box that minimize the amount of material used?
Just from $13/Page
Order Essay

The amount of material used is directly proportional to the surface area, so we will minimize the amount of material by minimizing the surface area.

The surface area of the box described is ##A=x^2 +4xh##

We need ##A## as a function of ##x## alone, so we’ll use the fact that
##V=x^2h = 32,000## cm^3

which gives us ##h = (32,000)/x^2##, so the area becomes:

##A=x^2 +4x((32,000)/x^2) = x^2 +(128,000)/x##

We want to minimize ##A##, so

##A’ = 2x-(128,000)/x^2 = 0## when ##(2x^3-128,000)/x^2 = 0##

Which occurs when ##x^3 – 64,000 = 0## or ##x=40##

The only critical number is ##x=40## cm.

The second derivative test verifies that ##A## has a minimum at this critical number:
##A” = 2+(256,000)/x^3## which is positive at ##x = 40##.

The box should have base 40 cm by 40 cm and height 20 cm.

(use ##h = (32,000)/x^2## and ##x=40##)

Show more



[ad_2]

Source link

Place your order
(550 words)

Approximate price: $22

Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
$26
The price is based on these factors:
Academic level
Number of pages
Urgency
Basic features
  • Free title page and bibliography
  • Unlimited revisions
  • Plagiarism-free guarantee
  • Money-back guarantee
  • 24/7 support
On-demand options
  • Writer’s samples
  • Part-by-part delivery
  • Overnight delivery
  • Copies of used sources
  • Expert Proofreading
Paper format
  • 275 words per page
  • 12 pt Arial/Times New Roman
  • Double line spacing
  • Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read more

Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read more

Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read more

Privacy policy

Your email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read more

Fair-cooperation guarantee

By sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more

Order your essay today and save 15% with the discount code BANANA