Answer:
Width = 5.5 ft.
Length = 18 ft.
Explanation:
Relating the length to the width:
Since the length (L) is 7 ft. more than double the width (W), we can represent this with the following equation:
L = 2W + 7
Formula for the area of a rectangle:
The formula for the area of a rectangle is given by:
A = LW, where
- A is the area in units squared,
Finding the width:
Thus, we can find the width (W) by substituting 99 for A and 2W + 7 for L in the area formula:
99 = (2W + 7)(W)
99 = 2W^2 + 7W
Converting the quadratic to standard form:
- We can solve for W using the quadratic formula.
Before we can use the quadratic formula, we need to convert 99 = 2W^2 + 7W to the standard form of a quadratic, whose general equation is given by:
ax^2 + bx + c = 0, where
- a, b, and c are constants.
Thus, we can convert to standard form by moving 2W^2 and 7W to the left-hand side:
(99 = 2W^2 + 7W) - 2W^2 - 7W
-2W^2 - 7W + 99 = 0
Thus, -2 is our a value, -7 is our b value, and 99 is our c value.
The quadratic formula:
The quadratic formula is given by:
x = (-b ± √b^2 - 4ac) / 2a, where
- x are the solutions (the ± indicates that it is possible to have two solutions to a quadratic since squaring a positive and negative number gives us a positive number).
Thus, we can find the width (W) by plug in -2 for a, -7 for b, and 99 for c in the quadratic formula:
W = (-(-7) ± √(-7)^2 - 4(-2)(99)) / 2(-2)
W = (7 ± √841) / -4)
W = (7 ± 29) / -4
Now we can start separating the positive and negative solution:
W = -7/4 + (-29/4) and W = -7/4 - (-29/4)
W = -7/4 - 29/4 and W = -7/4 + 29/4
W = -9 and W = 5.5
Since we can't have a negative dimension, the width is 5.5.
Finding the length (L):
We can now find the length (L) of the rectangle by substituting 99 for A and 5.5 for W in the rectangle area formula:
(99 = 5.5L) / 5.5.
18 = L
Thus, the length (L) of the rectangle is 18 ft.
We know that our answers are correct since double the width is 11 as 5.5 * 2 = 11 and 7 more than this is 18 since 11 + 7 = 18.