SOLUTION: The area of a rectangle with a perimeter of 120 units can be described by y = x (60 - x), where x represents the width of the rectangle and y represents the area. A farmer has 120

Question 1023080: The area of a rectangle with a perimeter of 120 units can be described by y = x (60 - x), where x represents the width of the rectangle and y represents the area. A farmer has 120 yards of fencing available and wishes to enclose a rectangular area. To the nearest five years, what width gives the largest enclosed area?

Found 2 solutions by josmiceli, solver91311:
Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!

This is a parabola with a maximum because
of the minus sign in front of the term
-------------------
The formula for x-coordinate of maximum is:

The width is 30 yds
-----------------
check:

yds2
-----------------
Try and
You should get less than yds2
also:
Let = perimeter

OK
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

To the nearest 5 years? What are you talking about?

Let a rectangle have a perimeter , then if the length is and the width is , we can say that , and conclude that

Since we know that the area of the rectangle is length times width, we can write an expression for area as a function of width by saying:

Since the vertex of the parabola is at the point where:

So, for the area function we derived above, the maximum value of the function is where:

Which is to say, you get the maximum area when the width is exactly one-fourth of the perimeter. The only way for the width to be exactly one-fourth of the perimeter is for the rectangle to have four equal measure sides, i.e. a square.

John

My calculator said it, I believe it, that settles it

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