Optimizing Silver Plating Costs for a Square Box and Athletics Track

[clairez93]

Homework Statement

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]

Homework Equations

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.

 

[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.

 

[whitay]

Question 1 is probably just a rounding error.

 

[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?

You should be minimizing your expression for the cost instead.

problem 2

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

general LaTeX pointers

to get your \pis 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 $

 

[whitay]

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?

You should be minimizing your expression for the cost instead.

problem 2

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

general LaTeX pointers

to get your \pis 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 $

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