A farmer has a rectangular garden plot surrounded by 200 ft of fence. Find the length and width of the garden if its area is 2400 ft2.

2.A rectangular garden is 2 ft longer than it is wide. Its area is 783 ft2. What are its dimensions?

To find the length and width of the garden in the first question, we can start by setting up a system of equations based on the given information.

Let's assume that the length of the garden is L and the width is W.

1. The perimeter of a rectangle is calculated by adding all four sides together. In this case, the perimeter is given as 200 ft, so we can write the equation:
2L + 2W = 200

2. The area of a rectangle is calculated by multiplying the length and width, which is given as 2400 ft², so we have the equation:
L * W = 2400

We now have a system of two equations with two variables. We can solve this system using various methods such as substitution or elimination.

Let's solve it using the substitution method:

From equation 1, we can isolate L:
2L = 200 - 2W
L = 100 - W

Substitute this value of L into equation 2:
(100 - W) * W = 2400

Expand and rearrange this equation:
- W² + 100W - 2400 = 0

Now we have a quadratic equation. We can solve it by factoring, using the quadratic formula, or by completing the square. In this case, let's solve it by factoring:

(W - 40)(W + 60) = 0

This gives two possible values for W: W = 40 or W = -60.

Since a negative width doesn't make sense in this context, we can discard the negative solution.

If W = 40, then L = 100 - W = 100 - 40 = 60.
So, the length of the garden is 60 ft and the width is 40 ft.

Therefore, the length and width of the garden in the first question are 60 ft and 40 ft, respectively.

Moving on to the second question:

Let's assume that the width of the garden is W.

1. The length of the garden is given as 2 ft longer than the width, so the length is W + 2.

2. The area of the rectangle is given as 783 ft², so we have the equation:
(W + 2) * W = 783

Expand this equation:
W² + 2W = 783

Rearrange it to form a quadratic equation:
W² + 2W - 783 = 0

Using factoring, quadratic formula, or completing the square, we can find that the solutions are:
W = 27 or W ≈ -29.2

Again, we discard the negative solution since a negative width doesn't make sense in this context.

If W = 27, then the length is W + 2 = 27 + 2 = 29.

Therefore, the width and length of the garden in the second question are 27 ft and 29 ft, respectively.