Question

The length of a rectangle is 3x +1 ft. and the width is x - 5 ft. The area of the rectangle is 112 sq. feet. Find the length and the width.

a) Length = 8 ft, Width = 7 ft
b) Length = 12 ft, Width = 9 ft
c) Length = 10 ft, Width = 7 ft
d) Length = 16 ft, Width = 7 ft

Answer 1

Upon solving the quadratic equation derived from the area of the rectangle, we find that the length is 28 feet and the width is 4 feet, which is not represented in the options provided, indicating a possible error in the question or the options.

Step-by-step explanation:

To solve for the length x and width of the rectangle with an area of 112 sq. feet, where the length (L) is described by the expression 3x + 1 feet and the width (W) by x - 5 feet, we can set up an equation using the formula for the area of a rectangle, which is Area = Length × Width. Hence, our equation becomes (3x + 1)(x - 5) = 112.

Next, we will follow these steps:

1. Expand the equation: 3x^2 - 15x + x - 5 = 112.
2. Simplify the equation: 3x^2 - 14x - 5 - 112 = 0 which simplifies further to 3x^2 - 14x - 117 = 0.
3. Factor the quadratic equation: (x - 9)(3x + 13) = 0.
4. Solve for x: x = 9 or x = -13/3. Since the width cannot be negative, we ignore the negative solution.
5. Substitute x = 9 into both expressions: Length = 3(9) + 1 = 28 feet, Width = 9 - 5 = 4 feet.

We find that the length is 28 feet and the width is 4 feet, which is not one of the provided options, suggesting a potential error in the problem statement or the options given.