Three sides of a fence and an existing wall form a rectangular enclosure. The total length of a fence used for the three sides is 160 feet. Find the value(s) for which the area is 2800 square feet.
Let the length of the side parallel to the wall be x. The two perpendicular sides will then each be y in length.
Solve this pair of equations:
x + 2y = 160
x y = 2800
(160-2y)y = 2800
-2y^2 +160y = 2800
y^2 -80y +1400 = 0
Solve for y or factor. It looks to me like the answer is not an integer, there may be two solutions. Use the quadratic equation.
y = [80 + sqrt(800)]/2
To solve this problem, let's start by breaking down the information given:
1. The total length of the fence used for the three sides is 160 feet.
2. The sides of the enclosure consist of three fence sides and an existing wall.
3. We need to find the value(s) for which the area is 2800 square feet.
Let's assume the dimensions of the rectangular enclosure as follows:
Let x be the length of the wall, and y be the length of the fence used for the opposite side of the wall (forming the length of the enclosure).
Let z be the length of the fence used for the width of the enclosure.
From the given information, we can deduce the following equations:
1. x + y + z = 160 (the total length of the fence is 160 feet).
2. The area of a rectangle is given by A = length × width.
Since the total length of the fence used for the three sides is 160 feet, we can rewrite equation 1 as:
x + y + z = 160
Next, we need to express the area of the rectangle in terms of x, y, and z:
Area = xy
From the given information, we are told that the area is 2800 square feet. Therefore, the equation becomes:
xy = 2800
Now we have a system of equations:
x + y + z = 160
xy = 2800
We can solve this system of equations by substitution or elimination to find the values of x, y, and z.
Let's use substitution:
1. Solve the first equation for z:
z = 160 - x - y
2. Substitute the value of z in the second equation:
xy = 2800
Substitute z = 160 - x - y into the equation:
x(160 - x - y) = 2800
This equation can be simplified and rearranged:
160x - x^2 - xy = 2800
x^2 + xy - 160x + 2800 = 0
Now, we can use this quadratic equation to find the values for x. By solving the equation, we can determine the corresponding values of y and z.