A rectangular parking lot is 25 ft. ny 50 ft is increased on all 4 sides by the same amount. The area increased by 400 sq ft. What are the new dimensions of the parking lot?
(25+L)(50+L)=25*50+400
25*50+75L+L^2=25*50+400
L^2+75L-400=0
l=(-75+-sqrt(75^2+1600))/2=(-75+-85)/2=
= 5
new dimensions: 30*55
To find the new dimensions of the parking lot, we need to determine the amount by which each side is increased.
Let's assume the increase is "x" feet.
Since the increase is the same on all four sides, the new length will be the original length plus the increase on both sides, which gives us (50 + 2x) ft.
Similarly, the new width will be (25 + 2x) ft.
Now we can set up an equation based on the given information:
(50 + 2x) * (25 + 2x) = 25 * 50 + 400
Expanding both sides of the equation gives us:
1250 + 100x + 100x + 4x^2 = 1250 + 400
Simplifying the equation further:
4x^2 + 200x - 400 = 0
Dividing the entire equation by 4:
x^2 + 50x - 100 = 0
We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
Here, a = 1, b = 50, and c = -100. Plugging these values into the formula, we get:
x = (-50 ± sqrt(50^2 - 4(1)(-100))) / (2 * 1)
Simplifying further:
x = (-50 ± sqrt(2500 + 400)) / 2
x = (-50 ± sqrt(2900)) / 2
x = (-50 ± 53.85) / 2
Now we have two possible values for x:
x = (-50 + 53.85) / 2 = 1.925
or
x = (-50 - 53.85) / 2 = -51.925
The negative value is not meaningful in this context since we can't have a negative increase in length. Thus, the only valid solution is x = 1.925 feet.
Now substitute this value of x into the expressions for the new length and width:
New length = 50 + 2 * 1.925 = 53.85 feet
New width = 25 + 2 * 1.925 = 28.85 feet
So, the new dimensions of the parking lot are approximately 53.85 feet by 28.85 feet.