The rectangular floor of a
shed has a length 4 feet longer than its width, and its
area is 140 square feet. Let x be the width of the floor.
(a) Write a quadratic equation whose solution gives
the width of the floor.
(b) Solve this equation.
x(x+4) = 140
Hmm. 10*14 = 140, so x=10 would be a solution.
To solve this problem, let's first understand the given information. According to the problem, the rectangular floor of the shed has a length 4 feet longer than its width. Let's assume the width of the floor is x feet. Therefore, the length of the floor would be x + 4 feet.
Now, we are also given that the area of the floor is 140 square feet. The area of a rectangle is given by the formula:
Area = Length * Width
So, we can write the equation as:
140 = (x + 4) * x
Now, let's solve this quadratic equation to find the width (x) of the floor.
(a) Write a quadratic equation whose solution gives the width of the floor:
The equation is 140 = (x + 4) * x.
(b) Solve this equation:
Expanding the equation, we get:
140 = x^2 + 4x
Rearranging the equation to bring all terms to one side, we have:
x^2 + 4x - 140 = 0
To solve this equation, we can either factor it or use the quadratic formula. Since the equation does not factor easily, let's use the quadratic formula:
The quadratic formula states that for an equation ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation x^2 + 4x - 140 = 0, we have a = 1, b = 4, and c = -140. Substituting these values into the quadratic formula, we get:
x = (-4 ± √(4^2 - 4 * 1 * -140)) / (2 * 1)
Simplifying further,
x = (-4 ± √(16 + 560)) / 2
x = (-4 ± √576) / 2
x = (-4 ± 24) / 2
This gives two possible solutions for x:
x1 = (-4 + 24) / 2 = 20 / 2 = 10
x2 = (-4 - 24) / 2 = -28 / 2 = -14
However, in the context of this problem, width cannot be negative. Therefore, the width of the floor is x = 10 feet.