h = -16t^2 + 60t + 0 where h is the height in feet and t is the time in seconds. Using this equation, what is the maximum range the ball will reach? Round to the nearest hundredth.
To find the maximum range of the ball, we need to determine the value of t when the height, h, is equal to zero. This is because at that time, the ball will be on the ground and no longer in the air.
Given that the equation is h = -16t^2 + 60t + 0, we can set h to zero and solve for t:
0 = -16t^2 + 60t
To solve this quadratic equation, we can factor out a common term of -4t:
0 = -4t(4t - 15)
Setting each factor equal to zero, we get:
-4t = 0 or 4t - 15 = 0
Solving these equations, we find:
t = 0 or 4t = 15
t = 0 or t = 15/4
Since time cannot be negative in this context, we can conclude that the ball will hit the ground at t = 0 seconds and t = 15/4 seconds.
To find the maximum range, we need to find the time it takes for the ball to travel from t = 0 seconds to t = 15/4 seconds.
The time it takes for the ball to reach the maximum range is given by:
t_max range = t (when h = 0) - t (when h = 0)
t_max range = 15/4 - 0
t_max range = 15/4
Now, to find the maximum range, we can substitute this value of t into the equation for h:
h = -16t^2 + 60t + 0
h_max range = -16(15/4)^2 + 60(15/4)
h_max range ≈ -16(225/16) + 900/4
h_max range ≈ -225 + 900/4
h_max range ≈ 675/4
Rounding to the nearest hundredth, the maximum range the ball will reach is approximately 168.75 feet.