The area of a rectangular wall of a barn is 80 square feet. The length is 12 feet longer than twice its width. Find the length and width of the barn.
Step 1: Let's assume the width of the barn is x feet.
Step 2: According to the given information, the length is 12 feet longer than twice the width. So, the length can be represented as (2x + 12) feet.
Step 3: The area of a rectangle is given by the formula: Area = Length × Width.
Step 4: We can substitute the given area and the values of length and width into the formula to get an equation. In this case, the equation is: 80 = (2x + 12) × x.
Step 5: Simplify the equation: 80 = 2x^2 + 12x.
Step 6: Rearrange the equation to a quadratic form: 2x^2 + 12x - 80 = 0.
Step 7: Divide each term in the equation by 2 to simplify it: x^2 + 6x - 40 = 0.
Step 8: Factor the quadratic equation: (x + 10)(x - 4) = 0.
Step 9: Set each factor equal to zero and solve for x:
For x + 10 = 0, x = -10.
For x - 4 = 0, x = 4.
Step 10: Since the width cannot be negative, we ignore the value x = -10 and conclude that the width of the barn is 4 feet.
Step 11: Substitute the value of the width into the expression for length: Length = 2x + 12 = 2(4) + 12 = 8 + 12 = 20.
Step 12: The length of the barn is 20 feet and the width is 4 feet.
To solve this problem, we can set up a system of equations based on the given information. Let's use "L" to represent the length of the barn and "W" to represent the width.
From the problem statement, we have two pieces of information:
1. The area of the rectangular wall is 80 square feet, so we have the equation L * W = 80.
2. The length is 12 feet longer than twice the width, which can be written as L = 2W + 12.
We can substitute the value of L from the second equation into the first equation to eliminate L and solve for W:
(2W + 12) * W = 80
2W^2 + 12W = 80
2W^2 + 12W - 80 = 0
Now let's solve this quadratic equation for W.
We can factor out 2 from the equation:
2(W^2 + 6W - 40) = 0
Then we can factor the quadratic expression inside the parentheses:
2(W + 10)(W - 4) = 0
Setting each factor equal to zero gives us two potential solutions:
W + 10 = 0 -> W = -10
W - 4 = 0 -> W = 4
Since the width of a barn cannot be negative, we discard the first solution, W = -10. So, the width of the barn is 4 feet.
Now we can substitute this value back into the second equation to find the length:
L = 2W + 12
L = 2(4) + 12
L = 8 + 12
L = 20
Therefore, the length of the barn is 20 feet and the width is 4 feet.
L = 2w + 12
L * w = 80
Substitute 2w+12 for L in the second equation and solve for w. Insert that value into the first equation to solve for L. Check by putting both values into the second equation.