Estimate the value of the quantity.
-A hat is dropped from a hot air balloon. The hat falls faster and faster but its acceleration decreases over time due to air resistance. The acceleration is measured every second after the drop for 6 seconds. Find an upper estimate for the speed when $t = 8$ seconds.
$\begin{array}{c|c}
\begin{array}{c}
\text { Time } \
(\mathrm{sec})
\end{array} & \begin{array}{c}
\text { Acceleration } \
\left(\mathrm{ft} / \mathrm{sec}^{2}\right)
\end{array} \
\hline 0 & 32.00 \
1 & 16.96 \
2 & 8.99 \
3 & 4.76 \
4 & 2.52 \
5 & 1.34 \
6 & 0.71
\end{array}$

A) $35.29 \mathrm { ft } / \mathrm { sec }$
B) $192 \mathrm { ft } / \mathrm { sec }$
C) $66.58 \mathrm { ft } / \mathrm { sec }$
D) $32.05 \mathrm { ft } / \mathrm { sec }$

Question

Estimate the value of the quantity.
-A hat is dropped from a hot air balloon. The hat falls faster and faster but its acceleration decreases over time due to air resistance. The acceleration is measured every second after the drop for 6 seconds. Find an upper estimate for the speed when $t = 8$ seconds.
$\begin{array}{c|c}
\begin{array}{c}
\text { Time } \
(\mathrm{sec})
\end{array} & \begin{array}{c}
\text { Acceleration } \
\left(\mathrm{ft} / \mathrm{sec}^{2}\right)
\end{array} \
\hline 0 & 32.00 \
1 & 16.96 \
2 & 8.99 \
3 & 4.76 \
4 & 2.52 \
5 & 1.34 \
6 & 0.71
\end{array}$

A) $35.29 \mathrm { ft } / \mathrm { sec }$
B) $192 \mathrm { ft } / \mathrm { sec }$
C) $66.58 \mathrm { ft } / \mathrm { sec }$
D) $32.05 \mathrm { ft } / \mathrm { sec }$

Quizplus · Accepted Answer

To estimate the upper speed at $t = 8$ seconds, we can sum the accelerations given for the first 6 seconds and assume the acceleration for the next two seconds is negligible due to the decreasing trend. The sum of the accelerations from $t = 0$ to $t = 6$ seconds is $32.00 + 16.96 + 8.99 + 4.76 + 2.52 + 1.34 + 0.71 = 67.28 \, \text{ft/sec}^2$. Since acceleration decreases over time, assuming it becomes very small after 6 seconds, the total velocity at $t = 8$ seconds can be approximated by the sum of the accelerations up to $t = 6$ seconds, which gives an estimate close to the value in choice C, $66.58 \, \text{ft/sec}$. This method overestimates the speed because it assumes constant acceleration for each second interval, while in reality, the acceleration is decreasing throughout each interval.

Estimate the Value of the Quantity t=8t = 8t=8 Seconds A) 35.29ft/sec35.29 \mathrm { ft } / \mathrm { sec }35.29ft/sec

Estimate the Value of the Quantity $t = 8$ Seconds A) $35.29 \mathrm { ft } / \mathrm { sec }$