The area of a triangle wall on a barn is 160 square feet. Its length is 4 feet longer than twice the width. Find the length and width of the wall of the barn
To find the length and width of the wall of the barn, we can set up an equation based on the given information.
Let's assume the width of the wall is represented by 'w'.
According to the problem statement, the length is 4 feet longer than twice the width. So, the length can be represented as '2w + 4'.
Now we can use the formula for the area of a triangle:
Area = (base * height) / 2
In this case, the base can be considered as the width, and the height can be considered as the length.
Area = (w * (2w + 4)) / 2
We know that the area is given as 160 square feet. So we can set up the equation:
160 = (w * (2w + 4)) / 2
To solve this equation, we can start by multiplying both sides by 2 to eliminate the fraction:
320 = w * (2w + 4)
Distribute w to eliminate the parentheses:
320 = 2w^2 + 4w
Rearrange the equation to form a quadratic equation:
0 = 2w^2 + 4w - 320
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic equation, a = 2, b = 4, and c = -320. Plugging these values into the quadratic formula:
w = (-4 ± √(4^2 - 4 * 2 * -320)) / (2 * 2)
Simplifying further:
w = (-4 ± √(16 + 2560)) / 4
w = (-4 ± √(2576)) / 4
Now, we take the square root of 2576:
w = (-4 ± 50.96) / 4
Now we solve for w using both the positive and negative values:
w1 = (-4 + 50.96) / 4
w2 = (-4 - 50.96) / 4
Calculating both solutions:
w1 = 46.96 / 4 = 11.74
w2 = -54.96 / 4 = -13.74
Since width cannot be negative, we disregard w2.
So, the width of the wall is approximately 11.74 feet.
To find the length, we substitute the value of w into our expression for the length: 2w + 4.
Length = 2 * 11.74 + 4
Length = 23.48 + 4
Length = 27.48 feet
Therefore, the width of the wall is approximately 11.74 feet, and the length is approximately 27.48 feet.