A storage box with a square base must have a volume of 80...

A storage box with a square base must have a volume of 80 cubic centimeters. The top and bottom cost $\$ 0.20$ per square centimeter and the sides cost $\$ 0.10$ per square centimeter. Find the dimensions that will minimize cost.

Key Concepts

Optimization

The optimization process entails finding the dimensions that minimize the cost function while still satisfying the volume constraint. This typically involves rewriting the cost function with a single independent variable using the constraint and then finding its minimum through calculus techniques.

Calculus - Derivatives and Critical Points

This key concept involves computing the derivative of the cost function with respect to the chosen variable, setting the derivative equal to zero to identify critical points, and then determining which of these points corresponds to the minimum cost. This process is fundamental in solving constrained optimization problems efficiently.

Volume Constraint

This concept involves ensuring that the dimensions of the box satisfy the fixed volume requirement. In problems like this, a constraint relates the variables (for instance, base side length and height) such that their product conforms to a given volume, allowing one to express one variable in terms of another.

Surface Area and Cost Function

This refers to calculating the areas of the different faces of a three-dimensional object, and then forming a cost function where each face’s area is multiplied by its respective cost per square centimeter. Here, different parts of the box (top, bottom, and sides) incur different costs, which must be incorporated into the total cost function.

Transcript

00:02 A storage box with a square base must have a volume of 80 cubic centimeters.

00:08 The top and the bottom cost 20 cents per square centimeter, and the sides cost 10 cents per square centimeter.

00:16 We need to find the dimensions that will minimize the cost.

00:22 So first of all, we are given that the volume of the box must be 80 cubic centimeters.

00:38 If we draw this box, we can let the length equal x and also the width equal x, and then the height of the box will let equal y.

01:06 So the first thing we have to do is to come up with the cost function.

01:10 I'm going to call the cost function c.

01:14 We need to have the cost function equal to the total price of top and bottom plus total.

01:44 Total price of the size.

01:57 Substituting, we have 2 times x squared multiplied by the 20 cents, plus 4 times xy times the 10 cents.

02:35 So it would be nice to have this function in terms of one variable.

02:41 I would like to replace the y with something in terms of x.

02:49 We know that a 80 is equal to x squared y.

02:59 X squared y is simply the volume of the box.

03:04 So we can solve for y.

03:06 Y is equal to 80 divided by x squared.

03:16 We can take this and substitute it right in for the y, yielding to x squared times 0 .2 plus 4x, and then we're going to substitute the 80 divided by x squared for the y, and then multiplied by 0 .1.

03:55 So if we simplify this algebraically, we will get 2x squared times 0 .2 plus 32 divided by x.

04:22 I would like to write that last term in terms of the negative exponent.

04:36 So plus 32 times x to the minus 1...

Please give Ace some feedback