Explain how to solve the following story problem A rancher plans to use 160 yd of fencing to enclose a rectangular corral and to divide it into two parts by a fence parallel to the shorter sides of the corral. Find the dimensions of the corral if its area is 1000

To solve this story problem, you can follow these steps: 1. Define your variables: Let's call the length of the rectangular corral L (in yards) and the width W (in yards). 2. Set up equations based on the given information: - The rancher plans to use 160 yards of fencing. Fencing is used for the perimeter of the corral and the dividing fence. So, you can set up an equation for the perimeter: Perimeter = 2L + 2W + Length of Dividing Fence = 160 yards - The area of the corral is given as 1000 square yards. The area of a rectangle is given by the formula: Area = Length x Width You know the area is 1000, so you can write: 1000 = L x W - The dividing fence is parallel to the shorter sides of the corral. This means that the dividing fence's length is equal to the width of the corral (W). 3. Create a system of equations: Now, you have two equations based on the information given: Equation 1: 2L + 2W + W = 160 Equation 2: L x W = 1000 4. Solve the system of equations: You can use Equation 2 to express one variable in terms of the other. For example, you can express L as L = 1000 / W. Now, substitute this expression for L into Equation 1: 2(1000 / W) + 2W + W = 160 Simplify and solve for W: 2000 / W + 2W + W = 160 Multiply through by W to get rid of the fractions: 2000 + 2W^2 + W^2 = 160W Combine like terms: 3W^2 - 160W + 2000 = 0 Now, you have a quadratic equation in terms of W. You can solve it using the quadratic formula: W = [-b ± √(b² - 4ac)] / (2a) In this case, a = 3, b = -160, and c = 2000. Calculate the discriminant (the value inside the square root): Discriminant = b² - 4ac = (-160)² - 4(3)(2000) = 25600 - 24000 = 1600 Now, use the quadratic formula to find W: W = [160 ± √1600] / (2 * 3) W = [160 ± 40] / 6 Now, you have two potential values for W: W₁ = (160 + 40) / 6 = 200 / 6 = 33.33 (rounded to two decimal places) W₂ = (160 - 40) / 6 = 120 / 6 = 20 5. Calculate the corresponding values for L: You already have an expression for L in terms of W from Equation 2: L = 1000 / W Calculate L₁ and L₂ using the values of W: L₁ = 1000 / 33.33 ≈ 30 (rounded to the nearest whole number) L₂ = 1000 / 20 = 50 So, there are two possible sets of dimensions for the rectangular corral: 1. If the width (W) is approximately 33.33 yards, then the length (L) is approximately 30 yards. 2. If the width (W) is 20 yards, then the length (L) is 50 yards. These are the dimensions of the corral that satisfy the given conditions.