Sydney drives 10 mi at a certain rate and then drives 20 mi at a rate 5 mi/h faster than the initial rate. write expressions for the time along each part of the trip. add these times to write an equation for the total time in terms of the initial rate, t total (r).

From the given we know that Sydney drives 10 mi at a certain rate. Let us call this initial rate as r. We also know that Sydney then drives 20 mi at a rate 5 mi/h faster than the initial rate.

We know that the relationship between time, distance and rate is:

where t is the time, d is the distance and r is the rate. We also know that the distances given to us in the question are constants at 10 mi/h and 20 mi/h, thus, the time, t is a dependent of the rate variable, r only and this will make the above equation to be:

Thus, for the first part of the trip, the time, can be expressed as :

Likewise, for the next part of the trip, the time can be expressed by:

The above two equations represent the expressions for the time along each part of the trip.

Adding both these times to write an equation for the total time in terms of the initial rate, , we will get:

The above is the total time in terms of the initial rate.