11. A punter kicked the football into the air with an upward velocity of 62 ft/s. Its height h in feet after t seconds is given by the function h = −16t^2 + 62t + 2. What is the maximum height the ball reaches? How long will it take the football to reach the maximum height?

Question

11. A punter kicked the football into the air with an upward velocity of 62 ft/s. Its height h in feet after t seconds is given by the function h = −16t^2 + 62t + 2. What is the maximum height the ball reaches? How long will it take the football to reach the maximum height? 
A. 3.875 seconds; 65.06 feet
B. 1.94 seconds; 62.06 feet
C. 0.97 seconds; 47.09 feet
D. 3.875 seconds; 62.06 feet

GPT 3.5 · Answer

To find the maximum height the ball reaches, we need to find the vertex of the quadratic function h(t) = -16t^2 + 62t + 2.

The t-coordinate of the vertex of a quadratic function in the form h(t) = ax^2 + bx + c is given by t = -b / 2a.

In this case, a = -16 and b = 62, so t = -62 / 2*(-16) = -62 / -32 = 1.94 seconds.

To find the maximum height, substitute t = 1.94 seconds into the function h(t) = -16t^2 + 62t + 2:

h(1.94) = -16(1.94)^2 + 62(1.94) + 2
h(1.94) = -16(3.7636) + 120.68 + 2
h(1.94) = -60.2176 + 120.68 + 2
h(1.94) = 62.4624 feet

Therefore, the football reaches a maximum height of approximately 62.46 feet and it takes 1.94 seconds to reach that height.

So, the correct option is:

B. 1.94 seconds; 62.06 feet