# Calc homework optimization

1. Apr 13, 2004

### prinzessin04

is there a good website on how to do optimization online?? we learning this section now in our calc class but our teacher didn't really explain anything, he only did one example and told us the rest were all similar but i din't know where to even begin on some of then...

#2. A company must manufacture a closed rectangular box with a square base. The volume must be 1040 cubic inches. The top and the bottom squares are made of a material that costs 3 dollars per square inch. The vertical sides are made of a different material that costs 4 per sqare inch.
What is the minimal cost of a box of this type?

for this one V=lwd
v=l^2w
w=1040/l^2
so the v=4(l^2)*3(1040/l^2)
i know i take the derivative after this but where do i go from there??

#3. Find the rectangle of largest area that can be inscribed in a semicircle of diameter 199, assuming that one side of the rectangle lies on the diameter of the semicircle.
The largest possible area is?

i'm not even sure what this question is asking of me... so someone understand this???

2. Apr 13, 2004

### formulajoe

#2 do this

V = a * b * c
A = 2ab + 2ac +2bc
than using v = 1040, do 1040 = a*b*c, solve for c, and plug c into the A equations. take partial derivative dA/da, and partial derivative dA/db. lemme know if this works.

3. Apr 13, 2004

### turin

Why did you change from a capital V to a lower case v? Whince did you get that last equation (it doesn't seem to agree with the others)?

I think you should define another dependent variable, let's call it C (cost, or total cost). The cost is determined by certain surface areas, whereas the specification is for a certain volume. As is probably apparent to you, the cost can take on a range of values even for this constant volume. I will also define the specific cost for the cheaper material as c3 and the specific cost for the more expensive material as c4.

This gives an equation (using your variables):

C = c3 2 l2 + c4 4 w l

Notice the factor of 2 in the first term, since there is a top and a bottom, and the factor of 4 in the last term, since there are 4 sides.

Since you want to minimize the cost, you want to find the l and w that minimizes C. For this, you use calculus (after you get C as a function of one variable, either l or w). The value at which the first derivative of C is zero and the second derivative of C is positive is the magical value (at least, magical to the marketing dept).

You can make use of symmetry. Obviously, you want both corners to touch the semicircle. Then You can fold the semicircle into a quarter circle and maximize for half of the rectangle. (If you maximize the area of half of the rectangle in this manner, the symmetry ensures that the whole rectangle area will be maximized.)