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A rectangular picture frame is to enclose an area of 72 in $^{2}$. If the cost of the top and bottom is twice the cost of the sides, determine the frame's dimensions if the total cost is to be minimized. Justify your conclusion!

$$6 \times 12$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Baylor University

Idaho State University

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.

06:23

A box of volume $72 \mathr…

05:31

A box of volume 72 $\mathr…

02:48

The cost of producing a re…

The following problem. We have a rectangular picture frame that is to enclose an area of 72" squared. If the top and bottom has cost twice as much of the sides, we want to determine the frames dimensions. If the total cost is to be minimized. So what this is going to look like is the area Which is equal to link things with. We noticed 72" squared and then we know the cost is equal to the perimeter but adding in some cost to it, it can be two times the width and because we know it's top and bottom and that has its own cost And that's me plus four times Mhm. The length, because we see there is not only true length but it's twice the cost. So this is the cost right here. We can get l in terms of W for example, mhm And then um put this over here. So then when we do that, we end up being able to minimize this and we see that's going to be when W equals 12. So it's gonna be a six by 12 frames

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