Question

the height above the ground, measured in feet, of a ball thrown off a 85 ft cliff with an initial velocity of 9 feet per second is given by h(t)=85 + 9t - 16t². what is the error in measuring the height of the ball after 1.1 seconds of motion with an error in time measurement of 0.02 seconds? error in height = feet

Explanation:

Step1: First, find the actual height at $t = 1.1$ seconds

We have the height - function $h(t)=85 + 9t-16t^{2}$. Substitute $t = 1.1$ into the function:
\[

$$\begin{align*} h(1.1)&=85+9\times1.1 - 16\times(1.1)^{2}\\ &=85 + 9.9-16\times1.21\\ &=85 + 9.9 - 19.36\\ &=75.54 \end{align*}$$

\]

Step2: Then, find the height with the error in time

The time with error is $t'=1.1 + 0.02=1.12$ seconds. Substitute $t'$ into the height - function:
\[

$$\begin{align*} h(1.12)&=85+9\times1.12-16\times(1.12)^{2}\\ &=85 + 10.08-16\times1.2544\\ &=85 + 10.08-20.0704\\ &=75.0096 \end{align*}$$

\]

Step3: Calculate the error in height

The error in height $\Delta h=h(1.1)-h(1.12)$.
\[

$$\begin{align*} \Delta h&=75.54 - 75.0096\\ &=0.5304 \end{align*}$$

\]

Answer:

$0.5304$ feet