Question

You have $500$500 to build a rectangular fence to enclose a yard next to your house and divide the yard into two parts. The left and right sides cost $2$2 per foot, and the back side costs $5$5 per foot. The front side is along the house and costs nothing. The fence in the middle costs $1$1 per foot. What is the largest area that can be enclosed? (Your answer should be the area, not the values of xx and yy.)

          You have $500$500 to build a rectangular fence to
enclose a yard next
to your house and divide the yard into two parts. The left
and
right sides cost $2$2 per foot, and the back side
costs $5$5 per foot. The
front side is along the house and costs nothing. The fence in the
middle
costs $1$1 per foot. What is the largest area that can be
enclosed? (Your answer should be the area, not the values
of xx and yy.)

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Added by Steve B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Eduard Sanchez

02/20/2023

Step 1

Start by drawing a rectangle on the canvas with the width of the fence in the middle and the length of the fence on the sides. Show more…

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You have $500$500 to build a rectangular fence to enclose a yard next to your house and divide the yard into two parts. The left and right sides cost $2$2 per foot, and the back side costs $5$5 per foot. The front side is along the house and costs nothing. The fence in the middle costs $1$1 per foot. What is the largest area that can be enclosed? (Your answer should be the area, not the values of xx and yy.)

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Transcript

00:01 Hi there, so for this problem, the available fencing for this that corresponds to the perimeter is equal to 500.

00:12 Now, we're going to label the horizontal side of this as x and the vertical sides as y.

00:21 Then we know that the one in the middle also has a dimension x, okay? now, then we are given information about the cost for each side.

00:40 So with that we can write the cost function for this.

00:44 Okay, so the left and right sides cost $2 per foot.

00:50 So we need to multiply two times, and we have two sides, there are y.

00:56 So that will be then two times y in here.

00:59 This plus and the back side costs 5 per food.

01:05 So x, we will have 5 times x.

01:11 And the one in the middle, the femmes in the middle, costs 1 per food.

01:18 So that will be just simply x.

01:21 So the cost function becomes 4 times 1 plus 6 times x.

01:27 Now, let's write the expression for the perimeter, the perimeter is just a sum of all of the sides for this.

01:33 So we'll have 2 times y plus 2 times x.

01:38 Simplifying this, we can divide everything by 2.

01:42 So we'll have 250 is equal to x plus y.

01:46 Now in here we can solve for y, so that will be 250 minus x.

01:51 And then we can substitute that into the cost function...

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