Optimization Problems

1. Nov 29, 2008

clairez93

1. The problem statement, all variables and given/known data

1. A closed box of square base and volume 36 cm^3 is to be constructed and silver plated on the outside. Silver plating for the top and the base costs 40 cnets per cm^2 and silver plating for the sides costs 30 cents per cm^2. Calculat ethe cost of plating the box so that the cost is minimized.

2. An athletics track has two "straights" of length l meters and two semi-circular ends of radius x meters. The permieter of the track is 400 m. What values of l and x produce the largest area inside the track? [see attachment #5 for diagram]

2. Relevant equations

3. The attempt at a solution

1. $$V = l^{2}h = 36$$

$$SA = l^{2} + 2lh$$

$$h = 36/l^{2}$$

$$SA = l^{2} + 72l^{-1}$$

$$SA' = 2l - 72l^{-2} = 2l - 72/l^{2} = 0$$

$$2l^{3} - 72 = 0$$

$$2l^{3} = 72$$

$$l = 3.302, w = 3.302, h = 3.302$$

$$C = (2lw)(.40) + 2(lh)(.30) + 2(hw)(.30) = 21.82$$

Edit: Latex makes my final cost not clear, it says $21.82. Answer Key:$21.60

2. $$2\pix + 4x + 2l = 400$$

$$\pix^{2} + 2lx = A$$

$$2l = 400 - 2\pix - 4x$$

$$l = 200 - \pix - 2x$$

$$\pix^{2}+(400 - 2\pix - 4x)x = A$$

$$-\pix^{2} - 4x^{2} + 400x = A$$

$$A' = -2\pix - 8x + 400 = 0$$

$$(-2\pi - 8)x = -400$$

$$x = 28.005, l = 56.010$$

Answer Key: x = 63.662, l = 0

I'm not sure what I did wrong.

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2. Nov 29, 2008

whitay

The second one. If the Key Answer is the correct answer. Then I also I solved and achieved that answer.

Try,

Parameter = $$400 = 2 \pi r + 2l$$
Area = $$\pi r^{2} + 2rl$$

Note: r = x

and work from there.

My answer was $$r = \frac{200}{\pi}$$ and l = 0.

I didn't have a calculator. But R seems correct.

Last edited: Nov 29, 2008
3. Nov 29, 2008

whitay

Question 1 is probably just a rounding error.

4. Nov 29, 2008

gabbagabbahey

problem 1

I'm assuming that $SA$ is supposed to represent the total surface area of the box? If so, then shouldn't it be : $$SA = 2l^{2} + 4lh$$ since the box has a top and bottom and 4 sides?

More importantly, why are you setting the derivative of the surface area equal to zero; wouldn't that minimize/maximize the surface area, not the cost?

problem 2

why is there a $4x$ in your expression for the perimeter ?

general LaTeX pointers

to get your $\pi$s to show up properly, ou need to put a space between \pi and your next letter: \pi x^2 shows up as $\pi x^2$ but \pix^2 shows up as $\pix^2$

to display the $\$ sign use \$instead of$

5. Nov 29, 2008

whitay

Sorry. But yeah \times looks like the cross product.