Increase length by 27.02 feet and width by 8.88 feet per side.
The parking lot currently has an area of 40,500 square feet (270 x 150). To add 18,400 square feet, the new area will be 58,900 square feet.
Let's represent the increase in length on each side by "x" and the increase in width on each side by "y".
The equation for the new area is:
(270 + 2x) * (150 + 2y) = 58,900
Expanding the equation:
40,500 + 1080x + 900y + 4xy = 58,900
Combining like terms:
40,500 + 1080x + 900y = 18,400 + 4xy
Since we want to find x and y independently, we can rewrite the equation:
1080x + 900y = -4xy + 18,400 - 40,500
1080x + 900y = -4xy - 22,100
This equation is difficult to solve for x and y directly. We need to find a way to eliminate one of the variables.
One approach is to use the fact that the original area is 40,500 square feet:
270 * 150 = 40,500
Adding 2x to the length and 2y to the width, we get:
(270 + 2x) * (150 + 2y) = 40,500
270 * 150 + 540x + 300y + 4xy = 40,500
Combining like terms:
40,500 + 540x + 300y = 4xy
540x + 300y = -4xy
This equation looks similar to the previous one, but we can eliminate y by substituting 40,500 - 540x for y:
540x + 300(40,500 - 540x) = -4xy
540x + 12,150,000 - 162,000x = -4xy
Simplifying:
-108,000x + 12,150,000 = -4xy
Dividing both sides by -4x:
27,000 - 3,037.5y = y
Solving for y:
3,038.5y = 27,000
y = 8.88
Now that we know y, we can plug it back into the original equation to find x:
1080x + 900 * 8.88 = -4x * 8.88 + 18,400 - 40,500
1080x + 7,992 = -35.52x - 22,100
1115.52x = -30,092
x = 27.02
Therefore, the length needs to be increased by 27.02 feet on each side and the width needs to be increased by 8.88 feet on each side to add 18,400 square feet to the parking lot.