Thaddeus and lan start at the same location and drive in opposite directions, but leave at different times. when they are 365 miles apart, their combined travel is 16 hours. if thaddeus drives at a rate of 20 miles per hour and lan drives at a rate of 25 miles per hour, how long had each been driving?

thaddeus has been hours and lan has been hours.

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Step-by-step explanation:

If Thaddeus drives the whole 16 hours, the distance between them is ...

  distance = speed · time

  distance = 20 mi/h · 16 h

  distance = 320 miles.

It is 45 miles more than that. For each hour that Ian drives, their separation distance increases by (25 mph -20 mph)·(1 h) = 5 mi. Then Ian must have driven ...

  (45 mi)/(5 mi/h) = 9 h

The rest of the 16 hours is the time that Thaddeus drove: 7 hours.

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Let x represent the time Ian drives. Then 16-x is the time Thaddeus drives. Their total distance driven is ...

  distance = speed · time

  365 mi = (25 mi/h)(x) + (20 mi/h)(16 h -x)

  45 mi = (5 mi/h)(x) . . . . . . . . subtract 320 miles, collect terms

  (45 mi)/(5 mi/h) = x = 9 h . . . . . . divide by the coefficient of x

Comment on the solution

You may notice a similarity between the solution of this equation and the verbal discussion above. (That is intentional.) It works well to let a variable represent the amount of the highest contributor. Here, that is Ian's time, since he is driving at the fastest speed.

point a is the center of the circle that passes through points e, f, and g and the center of the circle that passes through points x, y, and z.