Afarmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. no fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. if the fencing costs $10 per linear foot to install and the farmer is not willing to spend more than $5000, find the dimensions for the plot that would enclose the most area. (width, length

width : 166.67 ft, length : 250 ft

Quadratic function is a function that has the term x²

The quadratic function forms a parabolic curve

The general formula is

where a, b, and c are real numbers and a ≠ 0.

The parabolic curve can be opened up or down determined from the value of a. If a is positive, the parabolic curve opens up and has a minimum value. If a is negative, the parabolic curve opens down and has a maximum value

So the maximum is if a <0 and the minimum if a> 0.

The formula for finding the coordinates of the maximum and minimum points of the quadratic function is the same.

The maximum / minimum point of the quadratic function is

Where

D = b²-4ac

Total length of fence :

x + 2y

Because the west side of the plot is shared with a neighbor, the equation becomes:

x + 1.5y

the fencing costs $ 10 per linear foot, and the farmer is not willing to spend more than $ 5000, then the equation becomes:

10x + 15y = 5000

x + 1.5y = 500 ---> x = 500-1.5y

The area of the square :

A = x.y

We input the value x

A = (500-1.5y) y

A = 500y-1.5y²

The maximum area of the quadratic function is (a <0 means the maximum value)

From the Area equation :

a = -1.5, b = 500, c = 0

D = b²-4ac

D = 500²-4.-1.5.0

D = 500²

D = 250,000

 Then the maximum area:

y = A: x

y = 41,666.6: 166.67

y = 250 ft

domain of the function

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the inverse of the ƒunction  

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F (x) = x2 + 1 g (x) = 5 - x

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step-by-step explanation:

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