Minimization: find dimensions that minimimze amount of wood

[f(x)'s]

Problem: Shane is making his girlfriend a wooden hinged jewellery box that is twice as long as it is wide. The volume of the box is to be 7200 cm^3. The are of the base must not exceed 900 cm^2 and the height must not exceed 15 cm. Determine the dimensions that minimize the amount of wood required to make the box.

I'm not sure how I can create a restriction for this problem because it can't exceed 900 or the height of 15... Although it's an optimization problem it's in the section of my quotient law and I can't figure out how it applies...

Advice pleeeeeeeeease!

 

[galactus]

I set up an equation:

length is twice width, so l=2w

The base will have dimensions \(\displaystyle (2w)w=2w^{2}\)

The sides dimensions: 2wh, because there are two we have \(\displaystyle 4wh\).

The sides on the ends dimensions: \(\displaystyle 2wh\)

\(\displaystyle \L\\S=2w^{2}+4wh+2wh\)[1]

\(\displaystyle \L\\h=\frac{7200}{2w^{2}}\)[2]

Sub [2] into [1] and get:

\(\displaystyle \L\\2w^{2}+\frac{21600}{w}\)

\(\displaystyle \L\\\frac{dS}{dw}=4w-\frac{21600}{w^{2}}\)

Solve for w:

\(\displaystyle \L\\w=6\cdot\5^{\frac{2}{3}}\approx{17.54}\)

Since the length is twice the width, we have Length = 35.08

\(\displaystyle h=\frac{7200}{2(17.54)^{2}}=11.69\)

base area: \(\displaystyle 2(17.54)^{2}=615.3\)

So far the criteria is met.

You have the dimensions, check the volume and see if it's 7200.

 

[f(x)'s]

Thank you - That was really helpful and I understand your steps!