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An aquarium is to be made in the shape of a rectangular solid with a square base and an open top. The volume of the aquarium is to be 108 cubic inches. What dimensions will minimize the amount of material needed to build it?

$$6 \times 6 \times 3$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

Missouri State University

Harvey Mudd College

University of Nottingham

Boston College

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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we're told that a. I'm an aquarium is to be made in the shape of a rectangle. Solid with a square base and an open top. The volume of the aquarium is to be 108 cubic inches. What dimensions will minimize the amount of material needed to build it. So here's our aquarium and top is open so the volume. Um Well regardless of the top being open or not, the volume is a square times age. So the base is a square. Um And then so that means a teak was wanna wait over a square. Now the surface area here is a square that's for the bottom And then there's 4 8 times eight sides. And since the top is open this this is just one and not two. Take this plugging into here. We get this for our surface area. Given this constraint, Take the derivative of this. Set a equals a one. So our optimum value and then I should have ones in here. And so we have to a one minus 4. 32. All over A one squared equals zero. And that tells us that a one after my value for a Is 6" and H one is 3". So we actually have a more of a very squat kind of, well maybe not that squad um aquarium which is kind of a weird, you know kind of a weird shape for an aquarium. But that's what we get if we want to minimize the surface area given the volume

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