The owner of a horse stable wishes to set up 4 rectangular corrals of identical dimensions along the back wall of an existing barn using 200 ft of fencing. The sides of each corral will be attached to the barn, fencing is not needed along the back wall. Find the function that expresses the combined area of 4 corrals each if each corral is x feet long.
Did you make a sketch ?
Even though you didn't say, I will assume that the corrals are joined with common lengths.
let the width of the entire corral (parallel to the barn) be y ft
let each width of corrals be x ft
So we have y + 5x = 200
y = 200 - 5x
Area = xy = x(200-5x) or 200x - 5x^2
The equation will vary depending on your definition of the variables. Mine avoids unnecessary fractions .
BTW, after you maximize the area function, the dimensions of each corral will be 20 by 25, for a maximum area of 2000 for the combined area
To find the function that expresses the combined area of the four corrals, we need to understand the given conditions and constraints.
Let's start by breaking down the problem and visualizing it. We know that:
- There are four corrals, each with identical dimensions.
- The sides of each corral will be attached to the barn.
- Fencing is not needed along the back wall.
- The total length of fencing available is 200 ft.
The combined area of the four corrals will consist of four rectangles placed side by side along the back wall of the barn. Let's assume each corral has a length of x feet and a width of y feet.
Now, let's consider the total amount of fencing required. Since three sides of each corral need fencing, and the corrals are adjacent, we can calculate the total length of fencing as follows:
Total fencing length = 4 × (2y + x) [4 corrals multiplied by the sum of two widths and one length]
According to the given information, the total length of fencing available is 200 ft, so we can set up an equation:
4 × (2y + x) = 200
Next, we want to find a function that expresses the combined area of the four corrals. The area of each corral is given by:
Area of one corral = x × y
The combined area of the four corrals will be:
Combined area = 4 × (x × y) = 4xy
Therefore, the function that expresses the combined area of the four corrals is:
Combined area = 4xy
To summarize, the function that expresses the combined area of the four corrals each with a length of x feet is 4xy, where x is the length of each corral and y is the width of each corral.