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A holding pen for fish is to be made in the form of a rectangular solid with a square base and open top. The base will be slate that costs $\$ 4$ per square foot and the sides will be glass that costs $\$ 5$ per square foot. If the volume of the tank must be 50 cubic feet, what dimensions will minimize the cost of construction?

$$5 \times 5 \times 2$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

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Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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in this case we have um cost to minimize and the cost is about his based on the amount of material. But we have two different materials. So we have a holding pen for fish is made from in the form of rectangular solid with a square base and open to the base will be slight And that costs four cents For $4 $4 per square foot. And the sides will be glass. That costs $5 per square foot. So the base here is slate. And so we want to we want to, the cost then Is then four times a squared. So that's $4 times the area of the base and then five times four A. H. Because we have four sides here. So that gives us 28 times eight. So this is the total cost given the dimensions of our holding um holding pen. Now we want the volume to be uh 50 cubic feet. So a square times HS 50 or HS 50 over a square plug that into here. And we get the cost is four times 250 plus ace cubed all over eight. We can take the derivative of that said a quote a one that set it to zero and we get um eight a one cubed minus 1. 25. All over a one squared. That tells us that um a one solving for you know basically this zero that says a one is five and then that means H one is two. So we have again a you know kind of a squat looking um And for a whole fish holding pen, that's probably a little more reasonable, I suppose. Um and the total cost then would be $300 given the cost of the materials, so this would be the minimum cost Given a volume of 50 50 cubic feet.

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