Answer to Question #274821 in Calculus for Olivia

Question #274821

A rectangular lot adjacent to a highway is to be enclosed by a fence. If the fencing costs $2.50 per foot

along the highway and $1.5 per foot on the other sides, find the dimensions of the largest lot that can

be fenced off for $720.


1
Expert's answer
2021-12-03T13:37:52-0500

Let "x=" the length of a fence along the highway, "y=" the width of a rectangular lot.

Then

"2.5x+1.5(x+2y)=720"

"x=\\dfrac{720-3y}{4}"

The area of the rectangle is


"A=xy"

Substitute


"A=A(y)=(\\dfrac{720-3y}{4})y, 0<y<240"

Find the first derivative wit respect to "y"


"A'(y)=((\\dfrac{720-3y}{4})y)'=180-1.5y"

Find the critical number(s)


"A'(y)=0=>180-1.5y=0"

"y=120"

Critical number:"120."


"A(0)=0"

"A(240)=0"

"A(120)=(\\dfrac{720-3(120)}{4})(120)=10800"

The area has the absolute maximum with value of "10800" for "0<y<240"

at "y=120."


"x=\\dfrac{720-3(120)}{4}=90"

The length of a fence along the highway is "90 \\ ft," the width of a rectangular lot is "120\\ ft."


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