To find the dimensions of the rectangular deck, we can start by letting the width of the deck be represented by the variable x. Given that the length of the deck is 5 feet longer than the width, we can express the length as x + 5.
The area A of a rectangle is calculated using the formula:
A = length × width
Substituting in the expressions we have for the length and width, we get:
310 = (x + 5) × x
This expands to:
310 = x2 + 5x
Rearranging the equation gives us:
x2 + 5x – 310 = 0
Now, we can apply the quadratic formula to solve for x. The quadratic formula is:
x = (-b ± √(b2 – 4ac)) / 2a
In our case, a = 1, b = 5, and c = -310.
Calculating the discriminant:
b2 – 4ac = 52 – 4(1)(-310) = 25 + 1240 = 1265
Now applying the quadratic formula:
x = (-5 ± √1265) / 2
Calculating the square root and simplifying:
√1265 ≈ 35.5, so:
x = (-5 ± 35.5) / 2
We have two potential solutions:
x = (30.5) / 2 ≈ 15.25
and
x = (-40.5) / 2 ≈ -20.25 (not a valid width)
Thus, the width of the deck is approximately 15.25 feet. Now, to find the length, we substitute back into the expression for length:
length = x + 5 = 15.25 + 5 = 20.25 feet
So, the dimensions of the rectangular deck are:
- Width: 15.25 feet
- Length: 20.25 feet
We can verify this by calculating the area:
Area = length × width = 20.25 × 15.25 ≈ 310 square feet, which matches the original condition.
Therefore, the dimensions of the rectangular deck are:
- Width: 15.25 feet
- Length: 20.25 feet