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?
To find the maximum height the ball reaches, we need to find the vertex of the parabolic function. This can be done by finding the time (t) at which the height (h) is maximum.
The height function is given by ℎ = −16t^2 + 62t + 2
To find the time at which the height is maximum, we need to find the vertex of the parabola.
The t-coordinate of the vertex is given by:
t = -b/2a where a = -16 and b = 62
t = -62 / (2*(-16))
t = -62 / -32
t = 1.9375 seconds
Now, we can substitute this time into the height function to find the maximum height:
h(1.9375) = -16(1.9375)^2 + 62(1.9375) + 2
h(1.9375) = -16(3.754) + 62(1.9375) + 2
h(1.9375) = -60.064 + 120.656 + 2
h(1.9375) = 62.592
Therefore, the maximum height the ball reaches is 62.592 feet.
To find how long it will take for the football to reach the maximum height, we already found the time to be 1.9375 seconds.