Lacy has $600 set aside to build a rectangular exercise kennel for her dogs. She will buy fencing material for $15/ft. Because the side of an existing barn will be used for one of the sides of the kennel, only three sides need to be fenced.

A.) Determine the dimensions of the kennel that will enclose the maximum area.
B.) What area will be enclosed?
If someone could please help me step by step with this question that would be awesome. TIA

To determine the dimensions of the kennel that will enclose the maximum area, we can follow these steps:

Step 1: Let's assume the length of the kennel is x feet. Since the barn side will be one of the sides, the width will be variable and we'll call it y feet.

Step 2: Since there are three sides to be fenced, the total length of fencing needed is x + 2y.

Step 3: According to the given information, Lacy has $600 set aside for the kennel, and the fencing material costs $15 per foot. Therefore, the cost of the fencing must not exceed $600, which can be expressed as:

15 * (x + 2y) ≤ 600

Step 4: Next, we need to express the area of the kennel in terms of x and y. The area of a rectangle is given by length multiplied by width, which in this case is x * y.

Step 5: We want to find the dimensions that will maximize the area, so that means we need to find the maximum value of the function x * y.

Now let's solve parts A and B of the question step by step:

A.) Determine the dimensions of the kennel that will enclose the maximum area.

To find the dimensions that maximize the area, we need to express the area in terms of a single variable. Let's do this by solving inequality (Step 3) for y:

15 * (x + 2y) ≤ 600
x + 2y ≤ 40 (divide both sides by 15)

Now, express y in terms of x:

2y ≤ 40 - x
y ≤ (40 - x)/2

Since we want to maximize the area, we will need to replace y in the area expression with (40-x)/2:

Area = x * y
Area = x * ((40 - x)/2)
Area = 20x - (x^2)/2

Now we have expressed the area in terms of x only. To find the dimensions that maximize the area, we can find the value of x that maximizes this function.

B.) What area will be enclosed?

To find the maximum area, we need to find the value of x that maximizes the function 20x - (x^2)/2. This can be done by finding the vertex of the quadratic equation x^2 - 20x + 40:

The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a = -1/2 and b = -20.

Substituting the values:
x = -(-20) / (2 * (-1/2))
x = 20

Now that we have the value of x, we can substitute it back into the area expression to find the corresponding value of y:

y = (40 - x)/2
y = (40 - 20)/2
y = 20/2
y = 10

So, the dimensions of the kennel that will enclose the maximum area are x = 20 feet (length) and y = 10 feet (width).

To find the area enclosed by the kennel, substitute the values of x and y into the area expression:

Area = x * y
Area = 20 * 10
Area = 200 square feet

Therefore, the area enclosed by the kennel will be 200 square feet.

A square will enclose the maximum area.

She has enough money for 40 feet of fencing.

I suggest dimensions of 14 by 13.