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## 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. [tex]V = l^{2}h = 36[/tex]

[tex]SA = l^{2} + 2lh[/tex]

[tex]h = 36/l^{2} [/tex]

[tex]SA = l^{2} + 72l^{-1}[/tex]

[tex]SA' = 2l - 72l^{-2} = 2l - 72/l^{2} = 0[/tex]

[tex]2l^{3} - 72 = 0 [/tex]

[tex]2l^{3} = 72[/tex]

[tex]l = 3.302, w = 3.302, h = 3.302[/tex]

[tex]C = (2lw)(.40) + 2(lh)(.30) + 2(hw)(.30) = $21.82[/tex]

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

Answer Key: $21.60

2. [tex]2\pix + 4x + 2l = 400[/tex]

[tex]\pix^{2} + 2lx = A[/tex]

[tex]2l = 400 - 2\pix - 4x[/tex]

[tex]l = 200 - \pix - 2x[/tex]

[tex]\pix^{2}+(400 - 2\pix - 4x)x = A[/tex]

[tex]-\pix^{2} - 4x^{2} + 400x = A[/tex]

[tex]A' = -2\pix - 8x + 400 = 0[/tex]

[tex](-2\pi - 8)x = -400[/tex]

[tex] x = 28.005, l = 56.010[/tex]

Answer Key: x = 63.662, l = 0

I'm not sure what I did wrong.