The height, s, in feet, of a rock thrown upward at an initial speed of 76 ft/s from a cliff 40 ft above the ocean beach is given by the function where t is the time in seconds. Find the maximum height above the beach that the rock will attain.
A) 2.4 ft
B) 122.25 ft
C) 130.25 ft
D) 134.25 ft
E) 130 ft

Question

The height, s, in feet, of a rock thrown upward at an initial speed of 76 ft/s from a cliff 40 ft above the ocean beach is given by the function  where t is the time in seconds. Find the maximum height above the beach that the rock will attain.
A) 2.4 ft 
B) 122.25 ft 
C) 130.25 ft 
D) 134.25 ft 
E) 130 ft

Quizplus · Accepted Answer

We need to find the maximum height attained by the rock, which means we need to find the vertex of the parabola given by the function. The formula for the vertex of a parabola with equation y = ax^2 + bx + c is (-b/2a, c - b^2/4a).

In this case, the equation of the function is

s(t) = -16t^2 + 76t + 40.

Therefore, a = -16, b = 76, and c = 40.

The time t at which the maximum height is attained is given by the formula -b/2a:

t = -b/2a = -76/(2*(-16)) = 2.375.

To find the maximum height, we substitute this value of t into the equation:

s(2.375) = -16(2.375)^2 + 76(2.375) + 40 ≈ 130.25.

Therefore, the maximum height attained by the rock is approximately 130.25 feet, which corresponds to choice C.

The Height, S, in Feet, of a Rock Thrown Upward