From time t = 0 to t = 4 an object's speed is given by the function $s ( t ) = t e ^ { 4 t }$ .Compute the distance travelled by the object during this time interval.Round your answer to two decimals.
A)8,330,724.81
B)8,330,728.68
C)8,330,738.94
D)8,330,830.01

Question

Quizplus · Accepted Answer

To find the distance traveled by the object from $t = 0$ to $t = 4$, we need to integrate the speed function $s(t) = te^{4t}$ over this interval. The integral of $s(t)$ with respect to $t$ gives us the distance traveled.$$\text{Distance} = \int_{0}^{4} te^{4t} dt$$To solve this integral, we use integration by parts, where we let $u = t$ and $dv = e^{4t}dt$, which gives $du = dt$ and $v = \frac{1}{4}e^{4t}$.$$\begin{align*}\text{Distance} &= \left. \frac{1}{4}te^{4t} \right|_0^4 - \int_{0}^{4} \frac{1}{4}e^{4t} dt \&= \frac{1}{4} \cdot 4e^{16} - \left. \frac{1}{16}e^{4t} \right|_0^4 \&= e^{16} - \frac{1}{16}e^{16} + \frac{1}{16} \&= \frac{15}{16}e^{16} + \frac{1}{16} \&\approx 8,330,728.68\end{align*}$$Therefore, the distance traveled by the object during the time interval from $t = 0$ to $t = 4$ is approximately 8,330,728.68, which corresponds to choice B.

From Time T = 0 to T = 4 an Object's