A prototype rocket takes off from a launchpad on the ground and is initially at rest.
The rocket has a time of flight of $T$ seconds.
The velocity of the rocket, $v \mathrm{~ms}^{-1}$, is given by
\[
v(t)=0.5 e^{t} \sin \left(\frac{\pi t}{a}\right)
\]
where $t$ is the time in seconds for $0 \leq t \leq T$ and $a$ is your assigned value in the table above.
(a) The rocket malfunctions and begins to descend towards the ground.
2
Find the time at which the rocket begins its descent.
(b) Use an approximation method with a suitable number of subintervals to estimate the
2 total distance travelled by the rocket during the first 10 seconds of flight.

Step 1 ：\(v(t) = 0.5 e^t \sin\left(\frac{\pi t}{a}\right)\)

Step 2 ：To find when the rocket begins its descent, we need to find when its velocity is 0. So, we set \(v(t)\) to 0:

Step 3 ：\(0 = 0.5 e^t \sin\left(\frac{\pi t}{a}\right)\)

Step 4 ：Since \(e^t\) is never 0, we can focus on the sine term:

Step 5 ：\(0 = \sin\left(\frac{\pi t}{a}\right)\)

Step 6 ：The sine function is 0 when its argument is a multiple of \(\pi\). So, we have:

Step 7 ：\(\frac{\pi t}{a} = n\pi\) for some integer n

Step 8 ：Solving for t, we get:

Step 9 ：\(t = na\)

Step 10 ：Since the rocket starts at rest and begins to descend, we choose the smallest positive integer value for n, which is 1:

Step 11 ：\(t = a\)

Step 12 ：So, the rocket begins its descent at \(\boxed{t = a}\) seconds.

Step 13 ：For part (b), we need to estimate the total distance traveled by the rocket during the first 10 seconds of flight. We can use the velocity function to find the distance traveled in each subinterval and sum them up.

Step 14 ：Let's use 10 subintervals of 1 second each. The distance traveled in each subinterval can be approximated by the average velocity in that subinterval multiplied by the length of the subinterval (1 second).

Step 15 ：For the i-th subinterval, the average velocity is:

Step 16 ：\(\frac{v(i-1) + v(i)}{2}\)

Step 17 ：So, the total distance traveled during the first 10 seconds is approximately:

Step 18 ：\(\sum_{i=1}^{10} \frac{v(i-1) + v(i)}{2}\)

Step 19 ：Now, we can plug in the velocity function and compute the sum:

Step 20 ：\(\sum_{i=1}^{10} \frac{0.5 e^{i-1} \sin\left(\frac{\pi (i-1)}{a}\right) + 0.5 e^i \sin\left(\frac{\pi i}{a}\right)}{2}\)

Step 21 ：After calculating the sum, we get the total distance traveled by the rocket during the first 10 seconds of flight.

Step 22 ：The final answer should be in the form of a decimal rounded to the nearest tenth.