1.Ruth has 240 feet of fencing available to enclose a rectangular field. Express the area A of the rectangular field as a function x, where x is the length of the field.

2. For what value of x is the area largest?

3. What is the maximum area?

let the length be x

let the width be w
but 2x + 2w = 240
x + w = 120
w = 120-x

area = xw = x(120-x)
= -x^2 + 120x

d(area)/dx = -2x + 120
= 0 for a max of area
-2x = -120
x = 60
then w = 120-60 = 60

for the max area the rectangle should be a square of 60 ft by 60 ft
for a max area of 3600 ft^2

1. Well, Ruth, it seems you've found yourself in quite a fencing situation. Now, let's get down to business. To express the area (A) of the rectangular field as a function of x, we need to work with the given information. Since you have 240 feet of fencing available, we can use it to enclose all four sides of the rectangular field.

The perimeter (P) of a rectangle is given by the formula P = 2(length + width). In this case, the perimeter (P) is equal to 240, so we can write the equation as 240 = 2(L + W).

Since we're interested in expressing the area (A) as a function of x (the length of the field), we want to solve for the width in terms of x. We rearrange the equation to get: W = 120 - x.

Now, we can substitute the value of W into the formula for the area of a rectangle (A = L * W): A = x(120 - x). And there you have it - the area A of the rectangular field as a function of x!

2. Ah, the question of finding the largest area. Well, Ruth, it's time to break out those mathematical skills! To find the value of x that gives us the largest area, we can turn to calculus. We need to find the critical points of the function A = x(120 - x), where the derivative equals zero.

Taking the derivative of A with respect to x, we get: A' = 120 - 2x. Setting this derivative equal to zero, we get 120 - 2x = 0. Solving for x, we find that x = 60.

3. The moment we've all been waiting for - the maximum area! To find the maximum area, we substitute the value of x we found (x = 60) into the equation for A: A = 60(120 - 60) = 60 * 60 = 3600 square feet. So, Ruth, the maximum area you can enclose with 240 feet of fencing is 3600 square feet. Enjoy your spacious rectangular field!

1. Let's start by defining the dimensions of the rectangular field. We'll assume the length of the field is x feet.

Since the perimeter of a rectangle is the sum of all its sides, we can write the equation:

2x + 2y = 240

Since the opposite sides of a rectangle are equal in length, the width, y, is also equal to x. Therefore, we can rewrite the equation as:

2x + 2x = 240

Simplifying further:

4x = 240

Dividing both sides by 4:

x = 60

Now we have the length of the field as x.

2. To find the value of x that gives us the largest area, we need to maximize the area function:

A = x * y

Since the width, y, is also equal to x, we can rewrite the area function as:

A = x * x

Simplifying further:

A = x^2

To find the maximum value of A, we need to find the vertex of the parabola formed by the graph of A = x^2. Since the coefficient of x^2 is positive, we know the parabola opens upwards.

Since the formula for the x-coordinate of the vertex of a parabola in the form y = ax^2 + bx + c is x = -b / (2a), the x-coordinate of the vertex is x = -0 / (2 * 1) = 0.

Therefore, the value of x that gives us the largest area is x = 0.

3. To find the maximum area, we substitute the value of x into the area function:

A = (0)^2

Simplifying further:

A = 0

Therefore, the maximum area is 0 square feet.

To answer these questions, let's break it down step by step:

1. Expressing the area A of the rectangular field as a function of x:
Remember that the formula to calculate the perimeter of a rectangle is P = 2l + 2w, where l is the length and w is the width. In this case, the perimeter is given as 240 feet. Since we're trying to find the area as a function of x, which represents the length of the field, we can write the equation for the perimeter as:
240 = 2x + 2w
Simplifying further, we can rearrange the equation to isolate w:
w = (240 - 2x) / 2
Now, the area of a rectangle can be calculated using the formula A = l * w. Substituting the expression for w into this equation, we get:
A(x) = x * ((240 - 2x) / 2)
Simplifying further, we have the function that expresses the area of the rectangular field, A(x), in terms of x.

2. Finding the value of x for which the area is largest:
To find the maximum area, we need to find the critical point of the function A(x). This can be done by taking the derivative of A(x) and setting it equal to zero. Let's calculate the derivative:
A'(x) = (dA/dx) = [x * (240 - 2x) / 2]' = (240 - 4x) / 2
Setting the derivative equal to zero:
(240 - 4x) / 2 = 0
Simplifying, we find:
240 - 4x = 0
Solving for x, we get:
x = 240 / 4 = 60
So the critical point is x = 60.

3. Finding the maximum area:
Now that we have the critical point, we can substitute it back into the function A(x) to find the maximum area:
A(60) = 60 * ((240 - 2 * 60) / 2)
Simplifying, we get:
A(60) = 60 * (120 / 2) = 60 * 60 = 3600 square feet
Therefore, the maximum area of the rectangular field is 3600 square feet.