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The perimeter of a rectangle is 24 feet.a. If $x$ is the measure of one side of the rectangle, represent the measure of an adjacent side in terms of $x .$b. If $y$ is the area of the rectangle, express the area in terms of $x$ .c. Draw the graph of the function written in b.d. What are the dimensions of the rectangle with the largest area?

a) $12-x$b) $y=x(12-x)$c) See graphd) 6 ft by 6 $\mathrm{ft}$

Algebra

Chapter 4

RELATIONS AND FUNCTIONS

Section 5

Polynomial Functions

An Introduction to Geometry

Functions

Linear Functions

Polynomials

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Okay, In this exercise, we are given that the perimeter of a rectangle is 24 feet. So that means if we add up the length of this side, this side, this side, this side get that the perimeter equals 24. So the first part of the question says, if X is the measure of one side of the rectangle expressed the length of the other side in terms of X. So let's think about how this breaks down. So if this is X, then this means this side length is also X. And we know that if we add them together and then add in the length of the other two sides, we're just gonna call those l we get plus two l equals 24. So we can actually, uh, find this l value in terms of X. So the first thing I'm gonna do is subtract two X from both sides of the equation. We have to l equals 24 minus two x. And if we divide away the two from every single term in the equation, we get l equals 12 minus X. Okay. So we can actually I'm gonna go ahead and erase all of that. That's not gonna work. Okay, We can actually rewrite l the quantity l as just being 12 minus x. Okay, so let's the first part of the question. The second part is, if why is the area of the rectangle express the area in terms of X? All right, So area, the area for a rectangle is given by base times height. So in this case, they're height is X, and our base, we just calculated is 12 minus X. So if why's the area well yet why equals x times the quantity 12 minus X. And if we expand that out in rearrange terms, we get negative X squared plus 12 x. Okay. And now we want a graph of this function, which I had already generated. But this is what the parable, a negative X squared plus 12 x looks like if you graph it on a set of axes and we need Thio for the last part of the question, find the dimensions of the largest rectangle. So this parable represents the area for each measurement value of X. So it looks like the area is at its greatest at 36 and that makes sense, because if we have a perimeter of 24 the highest area quadrilateral we can create with that is just a perfect square. And if we do that, we get that our area six times six is 36. If we were to do different permanent measures like, uh, 10 and wrong spot 10 and two, we would get that the area is 10 times two is 20. So, uh, 36 is clearly about as big as it gets and

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