A farmer wishes to fence off a rectangular field that
borders a straight river, as shown in the diagram
provided. He does not need to fence the side
bordering the river. The total length of the fencing
is 60 ft long.
a. Use your knowledge of quadratic functions and equations to find the dimensions of the
enclosure that will produce the maximum area.
b. What area will be enclosed?
let the width be x and the length be y
So you will need 2 widths and one length
2x + y = 60
y = 60-2x
area = xy = x(60-2x) = -2x^2 + 60x
At this point you have several options to find the vertex of this parabola, depending of your depth of study into this topic.
Judging by the wording of the question, I would guess you have learned the "completing the square" procedure.
Use it to find the vertex, which will tell you both answers.
To find the dimensions of the enclosure that will produce the maximum area, we can use the concept of quadratic functions and equations. Let's solve it step by step.
a. Let's assume the length of the field parallel to the river is x ft, and the width perpendicular to the river is y ft. We know that the total length of fencing is 60 ft. Since only three sides need to be fenced (the two sides and the width perpendicular to the river), the perimeter of the enclosure is:
Perimeter = x + 2y
Since the total length of the fencing is 60 ft, we have:
x + 2y = 60
Now, let's find an equation for the area of the enclosure. The area of a rectangle is given by:
Area = length * width
In this case, the area is:
Area = x * y
Since we want to find the dimensions that will produce the maximum area, we need to express the area in terms of a single variable so that we can differentiate it and find its maximum value.
Using the equation obtained earlier (x + 2y = 60), we can express x in terms of y:
x = 60 - 2y
Substituting this expression into the area equation, we get:
Area = (60 - 2y) * y
Simplifying, we have:
Area = 60y - 2y^2
Now we have the area equation in terms of a single variable, y. To find the maximum area, we differentiate the area equation with respect to y and set it equal to 0:
d(Area)/dy = 60 - 4y = 0
Solving this equation for y, we get:
60 - 4y = 0
4y = 60
y = 15
Now that we have the value of y, we can substitute it back into the equation x + 2y = 60 to find the value of x:
x + 2(15) = 60
x + 30 = 60
x = 30
Therefore, the dimensions of the enclosure that will produce the maximum area are x = 30 ft and y = 15 ft.
b. To find the area that will be enclosed, we can substitute the values of x and y into the area equation:
Area = x * y
Area = 30 * 15
Area = 450 ft²
Hence, the area that will be enclosed is 450 square feet.