To solve this problem, we’ll integrate the acceleration function to find the velocity $v(t)$ and then compute the distance traveled during the given time interval.

---

(a) Finding the Velocity $v(t)$:

The acceleration function is given as: $a(t) = t + 8$ To find the velocity function, integrate $a(t)$ with respect to $t$: $v(t) = \int a(t) \, dt = \int (t + 8) \, dt$

Perform the integration: $v(t) = \frac{t^2}{2} + 8t + C$

We use the initial condition $v(0) = 5$ to find $C$: $v(0) = \frac{(0)^2}{2} + 8(0) + C = 5 \quad \implies \quad C = 5$

Thus, the velocity function is: $v(t) = \frac{t^2}{2} + 8t + 5 \quad \text{(in m/s)}.$

---

(b) Finding the Distance Traveled:

The total distance traveled is the integral of the speed (magnitude of velocity) over the time interval $[0, 10]$.

The velocity function is: $v(t) = \frac{t^2}{2} + 8t + 5$

Since $v(t) \geq 0$ for $0 \leq t \leq 10$, the total distance traveled is the definite integral of $v(t)$: $\text{Distance} = \int_{0}^{10} v(t) \, dt = \int_{0}^{10} \left(\frac{t^2}{2} + 8t + 5\right) \, dt$

Compute the integral term by term: $\int \frac{t^2}{2} \, dt = \frac{t^3}{6}, \quad \int 8t \, dt = 4t^2, \quad \int 5 \, dt = 5t$

$\int_{0}^{10} v(t) \, dt = \left[\frac{t^3}{6} + 4t^2 + 5t \right]_0^{10}$

Evaluate at $t = 10$: $\text{At } t = 10: \quad \frac{(10)^3}{6} + 4(10)^2 + 5(10) = \frac{1000}{6} + 400 + 50 = 166.67 + 400 + 50 = 616.67 \, \text{m}.$

Evaluate at $t = 0$: $\text{At } t = 0: \quad \frac{(0)^3}{6} + 4(0)^2 + 5(0) = 0$

Thus, the total distance traveled is: $\text{Distance} = 616.67 \, \text{m}.$

---

Final Answers:

(a) $v(t) = \frac{t^2}{2} + 8t + 5 \, \text{m/s}$

(b) Distance traveled = $616.67 \, \text{m}$

---

Do you want detailed explanations for the steps?
Here are some follow-up questions for practice:

1. How would the velocity change if the initial velocity were $0 \, \text{m/s}$?
2. What is the velocity at $t = 10 \, \text{s}$?
3. How do you verify that $v(t) \geq 0$ on the given interval?
4. What would the distance be if $a(t) = t^2 + 8$?
5. Can you sketch the graph of $v(t)$ and interpret its shape?

Tip: When integrating to find displacement or velocity, always check initial conditions to determine constants of integration.