a. The area of the existing parking lot is 40,500 square feet.
b. The value of x is 2.
c. The length and width of the parking lot should be expanded by 20 feet on each side.
a. Find the area of the existing parking lot.
Area = 270 ft * 150 ft = 40,500 square feet
b. Write a verbal model and an equation that you can use to find the value of x.
The area of the new parking lot will be equal to the area of the existing parking lot plus the area of the expansion. We can represent the area of the expansion as (2x)(2x), where x is the distance by which the length and width of the parking lot are expanded.
Verbal model:
Area of new parking lot = Area of existing parking lot + Area of expansion
Equation:
(270 + 2x)(150 + 2x) = 40,500 + 18,400
c. Solve the equation from part (b) and find the value of x.
Expanding the left side of the equation, we get:
40,500 + 900x + 300x + 4x^2 = 58,900
4x^2 + 1200x - 18400 = 0
Dividing both sides of the equation by 4, we get:
x^2 + 300x - 4600 = 0
This quadratic equation can be factored as:
(x - 20)(x + 230) = 0
Therefore, x = 20 or x = -230. Since x represents the distance by which the length and width of the parking lot are expanded, we must take the positive solution.
Answer: The length and width of the parking lot should be expanded by 20 feet on each side.
Conclusion:
By expanding the length and width of the parking lot by 20 feet on each side, the science center can add 18,400 square feet to the area of the parking lot.