Question

1. no calculator

jacob is using his slingshot to throw peanuts out his bedroom window to feed the squirrels in the
woods behind his house. the height h(t) of the peanuts can be modeled by the quadratic function
$h(t)=-8t^{2}+20t + 14$ where height is measured in feet and time t is measured in seconds.
determine the average rate of change in the height of the peanut from 1 to 3 seconds.

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 \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 1\), \(b=3\), and \(f(t)=h(t)=-8t^{2}+20t + 14\).

Step2: Calculate \(h(3)\)

Substitute \(t = 3\) into \(h(t)\):
\(h(3)=-8\times(3)^{2}+20\times3 + 14=-8\times9+60 + 14=-72+60 + 14 = 2\)

Step3: Calculate \(h(1)\)

Substitute \(t = 1\) into \(h(t)\):
\(h(1)=-8\times(1)^{2}+20\times1 + 14=-8 + 20+14=26\)

Step4: Calculate the average rate of change

Using the formula \(\frac{h(3)-h(1)}{3 - 1}\), substitute \(h(3)=2\) and \(h(1)=26\):
\(\frac{2-26}{3 - 1}=\frac{-24}{2}=-12\)

Answer:

The average rate of change is \(-12\) feet per second.