Question

10. Recreational Planning A playground that is to be rectangular in shape has an

area modeled by the function f(x) = -x + 12x where x represent the length
of one side of the playground in meters.
a. What value of x gives the playground the greatest possible area?
b. What is the greatest possible area?

Answer 1

To find the value of x that gives the playground the greatest possible area, use the quadratic function f(x) = -x + 12x. The value of x that maximizes the area is 1/24 meters, and the greatest possible area is 11/6 square meters.

Step-by-step explanation:

To find the value of x that gives the playground the greatest possible area, we need to find the maximum value of the function f(x). In this case, the function is f(x) = -x + 12x. To find the maximum, we can find the vertex of the quadratic function.

The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a, b, and c are the coefficients of the quadratic function in the form ax^2 + bx + c. In this case, a = 12, b = -1, and c = 0. Plugging these values into the formula, we can find that x = 1/24.

Therefore, the value of x that gives the playground the greatest possible area is 1/24 meters. To find the greatest possible area, we can substitute this value of x back into the function f(x) = -x + 12x. Evaluating the function at x = 1/24 gives us the greatest possible area of 11/6 square meters.