Question

the height in feet, h, of a model rocket t seconds after launch is given by the equation h(t)=3 + 70t - 16t². 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:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). For the function \(h(t)=3 + 70t-16t^{2}\), \(a = 1\) and \(b = 3\).

Step2: Calculate \(h(1)\) and \(h(3)\)

• Substitute \(t = 1\) into \(h(t)\): \(h(1)=3+70\times1 - 16\times1^{2}=3 + 70-16=57\).
• Substitute \(t = 3\) into \(h(t)\): \(h(3)=3+70\times3-16\times3^{2}=3 + 210-144 = 69\).

Step3: Calculate the average rate of change

Using the formula \(\frac{h(3)-h(1)}{3 - 1}\), we substitute \(h(1) = 57\) and \(h(3)=69\). Then \(\frac{69 - 57}{3 - 1}=\frac{12}{2}=6\).

Answer:

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