Answer: The maximum height the football reaches = 51.89 feet.
Step-by-step explanation:
Given: The equation [tex]h = -16t^2+ 53t + 8[/tex] gives the height h, in feet, of a football as a function of time t, in seconds, after it is kicked.
Differentiate the given equation, we get
[tex]h'=-(2)16t+53=-32t+53[/tex] (i)
Put [tex]h'=0[/tex]
[tex]\Rightarrow\ -32t+53=0\\\\\Rightarrow\ t=\dfrac{53}{32}\approx1.65625[/tex]
Differentiate (i) w.r.t. t, we get
[tex]h''=-32[/tex] < 0 i.e. h is maximum at t= 1.65625
The value of h at t= 1.65625 will be [tex]-16(1.65625)^2+53(1.65625)+8=51.890625\approx51.89\text{ feet}[/tex]
Hence, the maximum height the football reaches = 51.89 feet.
