Pre-Calculus question

Question

You have 764 feet of fencing to enclose a rectangular plot of land with one side along a river. No fencing is to be used on the side of the plot along the river. Find the dimensions of the plot that will maximize the enclosed area. Enter the smaller dimension first.

Answer 1

Let x be the length of the each of the 2 identical sides and y be the length of the unique side.

The enclosed area is given by

A = xy

The perimeter equation

2x + y = 764 tells us that y = 764 – 2x

Replacing y with 764 – 2x in the area equation gives us the quadratic function

A(x) = x(764 – 2x) = –2x² + 764x

There are two methods to find the value of x that maximizes A.

Optimization Using Algebra

A(x) = –2x² + 764x is a quadratic equation with a negative leading coefficient (a = –2).

Because of this, the function A(x) describes a parabola the opens downward.

So its maximum value occurs at its vertex point.

The x-coordinate of the vertex of a parabola is given by –b/2a where a = –2 and b = 764.

So here the x-coordinate of the vertex is x = –764/2(–2) = 764/4 = 191, which is one of the dimensions we are looking for.

y = 764 – 2x = 764 – 2(191) = 382 is the other dimension.

So the dimensions of the plot that will maximize the enclosed area are

191 × 382 feet

(The formula –b/2a for the x-coordinate of the vertex comes from the method completing the square to convert the equation of the quadratic function to its vertex form.

Optimization Using Calculus

We start with the area function A(x) = –2x² + 764x obtained as described above.

If we take the derivative of A(x) and set it equal to zero, we can find the value of x where the slope of A(x) is zero, corresponding to the maximum or minimum of A(x).

dA/dx = –4x + 764 = 0 means that 4x = 764 and that x = 764/4 = 191

This value of x corresponds to the maximum, since the negative leading coefficient of the quadratic function tells us we have function describing a parabola that opens downward.

Answer 2

Let x = the length of the side parallel to the river.

Let y= the lengths of the sides that touch the river.

Then x + y + y = 764 feet

or x + 2y = 764 feet

x × y = area , in square feet.

Sloving the perimeter equation for x:

x = 764 -2y

So,

(764 -2y) × y = 764y -2y^2 = area

Since this is a quadratic the maximum (or minimum) value of the area occurs at y = -b/(2a).

a = -2 and b= 764, so

y = -764 / [ (-2) × 2 ] = 764/ 4 = 191 feet

764 -2×191 = 764 -382 = 383 feet.

And this will provide an area of 72,962 square feet.

Answer 3

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