1. ## geometric Optimization problem

A large bin for holding heavy material must be in the shape of a box with an open top and a square base. The base will cost 7 dollars a square foot and the sides will cost 9 dollars a foot. If the volume must be 170 cubic feet. Find the dimensions that will minimize the cost of the box's construction.

Base :
Each side:

Thank you!

2. Originally Posted by mistrz23
A large bin for holding heavy material must be in the shape of a box with an open top and a square base. The base will cost 7 dollars a square foot and the sides will cost 9 dollars a foot. If the volume must be 170 cubic feet. Find the dimensions that will minimize the cost of the box's construction.

Base :
Each side:

Thank you!
First you need to determine which two equations you will need to work your optimization problem. From the information given, it looks like we will use volume and surface area.

Let the height of the box = h
Let the width/length of the box = x

So,
$\displaystyle Volume = xh$
$\displaystyle Area = x^2 + 4xh$

We are given the volume and we are asked to find dimensions of the box. More specifically, the dimensions that will MINIMIZE the cost of building the box.

Plug the given volume $\displaystyle 170 ft^3$ into your volume equation and isolate your variable "h". Then substitute that into your area equation, so you will only have an equation in terms of x. Simplify. Take your derivative. To calculate the dimensions to minimize the cost, put your derivative = 0 and solve for x. Then plug your answer for x into the equation you used to substitute "h" and solve.

Then you can probably take it from here!