He needs two adjacent rectangular enclosures - he has 300 feet of fencing.

a. find a function that models the total area of enclosures with respect to the width of the enclosures.

b. write your function in vertex form using the vertex formula.

c. what is the largest area that farmer jack can enclose?

d. how should he use the fencing?

*use graphing calculator for a & b to verify*

To make the question "interesting", usually there is a common side for the two adjacent rectangles

Otherwise they would simply have to be 2 equal squares.

Did you have a diagram to show it ?

To solve this problem, we can break it down into several steps:

a. Finding a function that models the total area of the enclosures with respect to the width of the enclosures:

Let's assume the width of one enclosure is "x". Since there are two adjacent rectangular enclosures, the total width would be "2x".

The length of each enclosure would be (300 - 2x) to utilize all the fencing.

The area of each enclosure can be calculated as length multiplied by width, which gives us A = x(300 - 2x).

b. Writing the function in vertex form using the vertex formula:

The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.

To express our function in vertex form, let's first convert it to the standard form:
A = 300x - 2x^2

To determine the vertex, we know that the x-coordinate of the vertex (h) is given by the formula h = -b / 2a, where a = -2 and b = 300.
h = -300 / (2 * -2) = 75

Next, substitute the value of h into the function to determine the y-coordinate (k) of the vertex:
A = 300(75) - 2(75^2) = 11250

Therefore, the vertex form of the function is A = -2(x - 75)^2 + 11250.

c. Finding the largest area that Farmer Jack can enclose:

Using the function A = -2(x - 75)^2 + 11250, we can observe that the coefficient of the x^2 term (-2) is negative. This indicates that the parabola opens downward, and the vertex represents the maximum point.

Therefore, the largest area that Farmer Jack can enclose would be the y-coordinate of the vertex, which is 11250 square feet.

d. Determining how Farmer Jack should use the fencing:

To maximize the area, Farmer Jack should use the fencing to create enclosures with a width of 75 feet. This will result in each enclosure having a length of 150 feet.

Using a graphing calculator can help visualize the parabolic shape of the function and verify our calculations.