A rancher wishes to enclose a rectangular partitioned corral with 1932 feet of fencing. What dimensions of the corral would enclose the largest possible area?
To find the dimensions of the corral that would enclose the largest possible area, we can use the concept of optimization. Let's break down the problem into smaller steps:
1. Identify what we know:
- The corral is rectangular, so it has two sides of length "a" and two sides of length "b".
- The perimeter of the corral is given as 1932 feet.
- We need to find the values of "a" and "b" that maximize the area.
2. Write the equation for the perimeter:
- Since the perimeter is the sum of all sides, we can write: 2a + 2b = 1932.
3. Simplify the equation:
- Divide both sides by 2: a + b = 966.
4. Rearrange the equation:
- Solve for "b": b = 966 - a.
Now, we need to find the equation for the area:
5. Write the equation for the area of the corral:
- The area is equal to the product of the two dimensions (a * b): Area = a * (966 - a).
Since we want to find the largest possible area, we need to maximize this equation. We can use calculus to find the critical points.
6. Find the derivative of the area equation:
- d(Area)/da = 966 - 2a.
7. Set the derivative equal to zero and solve for "a":
- 966 - 2a = 0
- 2a = 966
- a = 483.
Now we can find the value of "b" using the equation we obtained earlier:
8. Substitute "a" into the equation b = 966 - a:
- b = 966 - 483
- b = 483.
So, the dimensions of the corral that would enclose the largest possible area are "483 feet" for both sides, making it a square corral.
To verify, we can calculate the area:
Area = a * b = 483 * 483 = 233,289 square feet.
Therefore, a corral with dimensions of 483 feet by 483 feet would enclose the largest possible area with 1932 feet of fencing.
A square would provide the largest area.
√1932 = 43.9545