### Video Transcript

A particle is moving in a straight line such that its speed at time 𝑡 seconds is given by 𝑣 equals 10𝑡 plus two meters per second, where 𝑡 is greater than or equal to zero. Given that its initial position, 𝑟 zero, is equal to 16 meters, find its position when 𝑡 equals three seconds.

The position at any given time to be calculated by integrating the velocity with respect to 𝑡. In this example, we need to integrate the expression 10𝑡 plus two. Integrating 10𝑡 gives us five 𝑡 squared. And integrating two give us two 𝑡. As there are no limits on our integral sign, we need to add the constant 𝑐. We were told that the initial position, 𝑟 zero, was equal to 16. Therefore, when 𝑡 equals zero, 𝑟 equals 16.

Substituting these values into the equation gives us 16 is equal to five multiplied by zero squared plus two multiplied by zero plus 𝑐. This means that our constant 𝑐 is equal to 16. The position of the particle at any time 𝑡 is given by five 𝑡 squared plus two 𝑡 plus 16.

As we were asked to find the position when 𝑡 equals three seconds. We need to substitute 𝑡 equals three into this equation. This gives us 𝑟 is equal to five multiplied by three squared plus two multiplied by three plus 16. Five multiplied by three squared is equal to 45. And two multiplied by three is equal to six. Adding 45, six, and sixteen gives us 67.

Therefore, the position when 𝑡 equals three seconds is 67 meters. We can use the equation 𝑟 equals five 𝑡 squared plus two 𝑡 plus 16 to find the position at any given time.