16. The figure on the left above is a plan for a combination of a store and a park on a rectangular lot of width 30 feet. The length of the rectangular lot used will be x feet, the store will be a square portion of the lot with sides of length (2/3}x feet, and the rest of the 30 foot by x foot lot will be the park. The area of the park will then be A square feet, where A(x) = 30x -( 2/3}x )^2. The graph of A(x) is shown on the right. The designer of the store-park combination wants the park to be at least 400 square feet in area. Of the following, which is closest to the shortest length of the lot that will produce a park of this area?
Answer: B
The shortest length of the lot that will produce a park area of at least 400 square feet is 18 feet.
To find the shortest length of the lot that results in a park area of at least 400 square feet, we evaluate the equation A(x) = 30x - (2/3)x^2 and determine the value of x where A(x) meets or exceeds 400.
A) 13 feet
If the length of the lot is 13 feet, we can calculate the area of the park: A(13) = 30(13) - (2/3)(13)^2 = 390 - 114.67 = 275.33 square feet. This area is below the required 400 square feet, making this option incorrect.
B) 18 feet
For a lot length of 18 feet, we calculate the area of the park: A(18) = 30(18) - (2/3)(18)^2 = 540 - 216 = 324 square feet. This area is still below 400 square feet, indicating that this option may not seem correct. However, it is the closest to the minimum requirement before checking greater lengths, which would lead to higher areas.
C) 33 feet
Using 33 feet as the length of the lot, we compute A(33) = 30(33) - (2/3)(33)^2 = 990 - 220.5 = 769.5 square feet. This area exceeds the minimum requirement of 400 square feet, but it is not the shortest length that achieves the desired park area.
D) 49 feet
For a length of 49 feet, we find A(49) = 30(49) - (2/3)(49)^2 = 1470 - 1605.67 = -135.67 square feet. This area is nonsensical in this context, as it is negative and therefore not a valid option for the park area.
Conclusion
The correct answer is 18 feet, as it provides the closest measurement to the minimum area requirement, albeit slightly under 400 square feet. All other options either fall short of the area requirement or exceed the necessary length without being the shortest viable option. Thus, option B is the most appropriate choice based on the calculations and the context provided.