Question

A farmer wishes to enclose a rectangular region bordering a river with fencing, as shown in the diagram. Suppose that $x$ represents the length of each of the three parallel pieces of fencing. She has $600 \mathrm{ft}$ of fencing available. (a) What is the length of the remaining piece of fencing in terms of $x$ ? (b) Determine a function $\mathscr{A}$ that represents the total area of the enclosed region. Give any restrictions on $x$. (c) What dimensions for the total enclosed region would give an area of $22,500 \mathrm{ft}^{2} ?$ (d) What is the maximum area that can be enclosed?

          A farmer wishes to enclose a rectangular region bordering a river with fencing, as shown in the diagram. Suppose that $x$ represents the length of each of the three parallel pieces of fencing. She has $600 \mathrm{ft}$ of fencing available.
(a) What is the length of the remaining piece of fencing in terms of $x$ ?
(b) Determine a function $\mathscr{A}$ that represents the total area of the enclosed region. Give any restrictions on $x$.
(c) What dimensions for the total enclosed region would give an area of $22,500 \mathrm{ft}^{2} ?$
(d) What is the maximum area that can be enclosed?

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Added by Elisa C.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

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Solved by Expert Donna Densmore

10/17/2022

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A farmer wishes to enclose a rectangular region bordering a river with fencing, as shown in the diagram. Suppose that $x$ represents the length of each of the three parallel pieces of fencing. She has $600 \mathrm{ft}$ of fencing available. (a) What is the length of the remaining piece of fencing in terms of $x$ ? (b) Determine a function $\mathscr{A}$ that represents the total area of the enclosed region. Give any restrictions on $x$. (c) What dimensions for the total enclosed region would give an area of $22,500 \mathrm{ft}^{2} ?$ (d) What is the maximum area that can be enclosed?

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Transcript

00:01 A farmer wants to enclose retain the region bordering a river.

00:06 It's going to have fencing.

00:08 Even though i can't see the diagram, i know this is going to be the fencing.

00:11 Okay, with three parallel pieces of fencing.

00:14 She has 600 feet of fence available.

00:17 So what's the length of the remaining piece of the fencing in terms of x? well, this is going to be x.

00:25 So we have four x's, and then so this is going to be a y.

00:28 So our perimeter, 600, is going to be 4 x plus y, which means y is going to be 600 minus 4x.

00:40 So what is going to be the parameters or what's the restrictions on x? well, first of all, it has to be greater than 0.

00:48 It can't equal 0 or we would not have any fence.

00:51 So it has to be greater than 0.

00:53 And then this has to be greater than 0 as well.

00:56 So 600 greater than 4x means we divide by 4.

01:00 We're going to have x.

01:02 It's got to be less than 150.

01:05 So zero is going to be somewhere between zero and 150.

01:10 Next we want to know the function for the area.

01:14 So we know area is length times width...

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