A diver jumps up and over from a cliff in order to avoid rocks and land in the deeper water. The path of the dive can be modeled using the function , where f(x) is the height, in feet, of the diver at x time in seconds. What is the maximum height, in feet, the diver will reach? Round your answer to the nearest hundredth if necessary. Do not include units in your answer.

Find the maximum height by solving for when the derivative of f(x) = 0 and the sign changes from positive to negative:

f(x) = -16x² + 2x + 48

Find the derivative of each term:

f'(x) = -32x + 2

Set the equation equal to 0:

0 = -32x + 2

-2 = -32x

x = 1/16 sec

Validate that the derivative changes from + to - at this value by plugging in values into the derivative formula on both sides of 1/16:

f'(0) = 2, +

f'(2) = -62, -

Therefore, there is a maximum at x = 1/16 because the function changes from positive to negative. Find the height at x = 16 seconds. (Plug it into the original equation)

f(1/16) = -16(1/16)² + 2(1/16) + 48

f(1/16) = -1/16 + 2/16 + 48

f(1/16) = 1/16 + 48

f(1/16) = 48 1/16 or 48.0625 ft

Round to nearest hundredth: ≈48.06 feet.

15 batches are the most you can make

step-by-step explanation: