Given that the distance traveled by the ball after t seconds is
[tex]s(t)=16t^2\text{ ft}[/tex](a) We have to find the distance traveled by the ball during the time interval [4,4.5]
Here, the time t1 = 4 sec
The time t2 = 4.5 sec
The distance traveled during this time interval will be
[tex]\begin{gathered} s(t2)-s(t1)=16(4.5)^2-16(4)^2 \\ =324-256 \\ =\text{ 68 ft} \end{gathered}[/tex]Thus, the ball traveled 68 ft during the time interval [4,4.5].
(b) We have to calculate the average velocity over the time t1 = 4 sec and
t2 = 4.5 sec.
The average velocity will be
[tex]\begin{gathered} v_{av}=\frac{16\times(4.5)^2-16\times(4)^2}{4.5-4} \\ =136\text{ ft/sec} \end{gathered}[/tex]Thus the average velocity is 136 ft/sec over time (4,4.5)
(c) We have to find the average velocity over the time intervals
[4.4.01], [4, 4.001],[4,4.0001],[3.9999,4],[3.999,4],[3.99,4]
The average velocity for the time interval [4,4.01] is
[tex]\begin{gathered} v_{av}1\text{ = }\frac{16\times(4.01)^2^{}-16\times(4)^2}{4.01-4} \\ =128.16\text{ ft/sec} \end{gathered}[/tex]The average velocity for the time interval [4,4.001] is
[tex]\begin{gathered} v_{av}2\text{ = }\frac{16\times(4.001)^2-16\times(4)^2}{4.001-4} \\ =128.016\text{ ft/sec} \end{gathered}[/tex]The average velocity for the time interval [4,4.0001] is
[tex]\begin{gathered} v_{av}3\text{ = }\frac{16\times(4.0001)^2-16\times(4)^2}{4.0001-4} \\ =128.0016\text{ ft/sec} \end{gathered}[/tex]The average velocity for the time interval [3.9999,4] is
[tex]\begin{gathered} v_{av}4\text{ = }\frac{16\times(4)^2-16\times(3.9999)^2}{4-3.9999} \\ =127.9984\text{ ft/sec} \end{gathered}[/tex]The average velocity for the time interval [3.999,4] is
[tex]\begin{gathered} v_{av}5\text{ = }\frac{16\times(4)^2-16\times(3.999)^2}{4-3.999} \\ =127.984\text{ ft/sec} \end{gathered}[/tex]The average velocity for the time interval [3.99,4] is
[tex]\begin{gathered} v_{av}6\text{ = }\frac{16\times(4)^2-16\times(3.99)^2}{4-3.99} \\ =127.84\text{ ft/sec} \end{gathered}[/tex]We can estimate the object's instantaneous velocity at t = 4 sec as
V(4) = 128 ft/sec
As all the values are nearing to 128 ft/sec
