Question

the height in feet, h, of a model rocket t seconds after launch is given by the equation ( h(t) = 3 + 70t - 16t^2 ). the average rate of change in ( h(t) ) between ( t = 1 ) second and ( t = 3 ) second is 6. what does the average rate of change tell you about the rocket?
the rocket is traveling six times as fast when ( t = 3 ) than it is when ( t = 1 ).
the rocket is at a greater height when ( t = 3 ) than it is when ( t = 1 ).
the rocket is 6 feet higher above the ground when ( t = 3 ) than it is when ( t = 1 ).
the rocket is traveling at a constant rate of 6 feet per second between ( t = 1 ) and ( t = 3 ).

Explanation:

1. Recall the definition of average rate of change: The average rate of change of a function \( y = h(t) \) between \( t = a \) and \( t = b \) is given by \( \frac{h(b)-h(a)}{b - a} \). This represents the average rate at which the function's value (in this case, height) changes with respect to time over the interval \([a,b]\).
2. Analyze each option:

• Option 1: The average rate of change is not a comparison of the speeds at individual points (\(t = 1\) and \(t=3\)) but an average over the interval. So this is incorrect.
• Option 2: The average rate of change of 6 does not mean the height at \(t = 3\) is 6 times greater than at \(t = 1\). To find the height difference, we calculate \(h(3)-h(1)\). Let's compute \(h(1)=3 + 70(1)-16(1)^2=3 + 70-16 = 57\) and \(h(3)=3+70(3)-16(3)^2=3 + 210-144 = 69\). The difference \(h(3)-h(1)=69 - 57=12\), not related to 6 times greater. So this is incorrect.
• Option 3: The height difference \(h(3)-h(1)=12\) (as calculated above), not 6. So this is incorrect.
• Option 4: By the definition of average rate of change, \( \frac{h(3)-h(1)}{3 - 1}=6 \) means that, on average, the height of the rocket (and thus its vertical speed, since height is a function of time) changes by 6 feet per second between \(t = 1\) and \(t = 3\). So this option correctly interprets the average rate of change.

Answer:

The rocket is traveling at a constant rate of 6 feet per second between \( t = 1 \) and \( t = 3 \).