# Math Help - Optimization questions

1. ## Optimization questions

1. U.S. parcel regulations state that packages must have length plus girth (girth equals the circumference of ends) of no more than 84 inches. Find the dimension of the cylindrical package of greatest volume that is mailable by post.

2. The manager of a department store wants to build a 600 square foot rectangular enclosure on the store's parking lot in order to display some equipment. Three sides of the enclosure will be built of redwood fencing at a cost of $14 per running foot. The fourth side will be built of cement blocks at a cost of$28 per running foot. Find the dimensions of enclosure that will minimize the total cost of building materials.

3. A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. If buildsing the tank costs $10 per sq. meter for the base and$5 per sq. meter for the sides, what is the cost of the least expensive tank?

Any help is appreciated! Thanks!

2. Originally Posted by live_laugh_luv27
1. U.S. parcel regulations state that packages must have length plus girth (girth equals the circumference of ends) of no more than 84 inches. Find the dimension of the cylindrical package of greatest volume that is mailable by post.

$2\pi r+h = 84$

$h = 84-2\pi r$

$V = \pi r^2 h$

$V = \pi r^2(84 - 2\pi r)$

distribute, simplify, find dV/dr and maximize

2. The manager of a department store wants to build a 600 square foot rectangular enclosure on the store's parking lot in order to display some equipment. Three sides of the enclosure will be built of redwood fencing at a cost of $14 per running foot. The fourth side will be built of cement blocks at a cost of$28 per running foot. Find the dimensions of enclosure that will minimize the total cost of building materials.

$LW = 600$

$L = \frac{600}{W}$

$C = 14(2L+W) + 28W$

$C = 14\left(\frac{1200}{W} + W\right) + 28W$

distribute, simplify, find dC\dW and minimize

3. A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. If buildsing the tank costs $10 per sq. meter for the base and$5 per sq. meter for the sides, what is the cost of the least expensive tank?

you set up the last one.
...

On 1, I'm stuck trying to find the dV/dr. I distributed and factored to v=2πr^2(42-πr). Is this correct so far?

For 2, I got length = 30, width = 20. Could you explain how you found the equation for the cost, please?

For 3, would I start with the formula for volume, length x width x height? How would I solve for a variable?