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Numerade Educator

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Problem 57 Hard Difficulty

Find a formula for the described function and state its domain.

A rectangle has perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides.

Answer

$10 L-L^{2}$ $0<L<10$

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Video Transcript

So in this problem, our goal is to find a formula for the area and to find the domain. And we should just start by sketching out what we know. So we have a rectangle we could label the sides with l for length and w for width. Now we know the perimeter is 20 and to get the perimeter, we would add the length and the width and the length and the width because we're going all the way around the rectangle. So we would have to Well, plus two w equals 20. And if we divide that by two, we have l plus w equals 10. Okay, Now, if we're going to find a formula for the area, remember that the area is length times width. However, that is only that is, in terms of two variables and we wanted Onley in terms of the length of one of its sides, so we need only one variable. So what we do is we go back to our perimeter equation and use that to find the value of W w equals 10 minus l. We can substitute that and for W in our area equation. And now we have area equals L times 10 minus l. And if we want to distribute, we get area as a function of l equals 10 l minus. L squared. So that's the formula that we're looking for now we want to consider the domain, so we don't know a whole lot about this rectangle. For all we know, it could be really, really, really long and very, very, very small on the with. Or it could have a very, very small link and a very, very big with. So we don't know exactly the relationship between those. But what we do know is that they must add to 10 lengthen with. So it could be that l is very, very, very close to 10. Or it could be that Ellis very, very, very close to zero. So then what we can say before the domain is that l is between zero and 10.