To find the distance traveled by an object over a given time interval when its velocity $v(t)$ is known, we need to compute the definite integral of the velocity function over that time interval.

Given the velocity function: $v(t) = 20 + 5 \cos(t)$

We want to find the distance traveled over the time interval from $t = 0$ to $t = 10$ seconds. The distance traveled is given by:

$\text{Distance} = \int_{0}^{10} v(t) \, dt = \int_{0}^{10} (20 + 5 \cos(t)) \, dt$

Step-by-Step Solution

1. Separate the Integral:

$\int_{0}^{10} (20 + 5 \cos(t)) \, dt = \int_{0}^{10} 20 \, dt + \int_{0}^{10} 5 \cos(t) \, dt$

2. Compute Each Integral:

• The first integral is straightforward:

[ \int_{0}^{10} 20 , dt = 20t \Big|_{0}^{10} = 20