To determine what the average rate of change tells us about the rocket, we first need to understand what this concept means.
The height [tex]\( h \)[/tex] of the rocket at any time [tex]\( t \)[/tex] is given by the equation:
[tex]\[ h(t) = 3 + 70t - 16t^2 \][/tex]
We are interested in the average rate of change of [tex]\( h(t) \)[/tex] between [tex]\( t = 1 \)[/tex] second and [tex]\( t = 3 \)[/tex] seconds.
1. Calculate the height at [tex]\( t = 1 \)[/tex] second:
[tex]\[ h(1) = 3 + 70(1) - 16(1)^2 \][/tex]
[tex]\[ h(1) = 3 + 70 - 16 \][/tex]
[tex]\[ h(1) = 57 \][/tex]
2. Calculate the height at [tex]\( t = 3 \)[/tex] seconds:
[tex]\[ h(3) = 3 + 70(3) - 16(3)^2 \][/tex]
[tex]\[ h(3) = 3 + 210 - 144 \][/tex]
[tex]\[ h(3) = 69 \][/tex]
3. Calculate the average rate of change:
The average rate of change of a function between two points [tex]\( t_1 \)[/tex] and [tex]\( t_3 \)[/tex] is given by:
[tex]\[ \text{Average rate of change} = \frac{h(t_3) - h(t_1)}{t_3 - t_1} \][/tex]
Substitute [tex]\( t_1 = 1 \)[/tex] and [tex]\( t_3 = 3 \)[/tex]:
[tex]\[ \text{Average rate of change} = \frac{69 - 57}{3 - 1} \][/tex]
[tex]\[ \text{Average rate of change} = \frac{12}{2} \][/tex]
[tex]\[ \text{Average rate of change} = 6 \][/tex]
The average rate of change being 6 means that, on average, the height of the rocket increases by 6 feet per second between [tex]\( t = 1 \)[/tex] second and [tex]\( t = 3 \)[/tex] seconds. Given this steady average increase, this tells us specifically that:
"The rocket is traveling at a constant rate of 6 feet per second between [tex]\( t=1 \)[/tex] and [tex]\( t=3 \)[/tex]."
