equation needed

Question

bill drives for 2 hours at 60 miles per hour and then drives 70 miles per hour for H hours. What is the equation to use to determine the average speed for his trip?

Answer 1

Speed is expressed as the ratio of distance over time.  It may be miles per hour (mi/hr), or yards per second (yd/sec), or ft/min, or kilometers/hour, or ...

 

To find the average speed, we often determine the total distance then divide by to total time.  Even though Bill did not actually drive this distance at this speed, if we drove the same distance as Bill did at the average speed at which Bill drove, we would arrive at the destination at exactly the same time as Bill did.  So, let's divide the total distance by the total time.

 

Now, remember:  D-I-R-T (Distance-Is-Rate-times-Time).    We will find 2 distances; the problem specifies 2 times.      (again,  Rate = Distance / Time)

 

Distance 1:   (2 hr)(60 mi/hr)  = 120 mi     (note: hr cancels out, leaving only mi)

Distance 2:   (H hr)(70 mi/hr) = 70H mi

Total Distance:     70H + 120

 

Time 1:  2 hr

Time 2:  H hr

Total Time:    H + 2 nr

 

Average Speed (Total distance / Total time):      (70 H + 120) / (H + 2)

 

Let's make up an easy problem:  Bill drives 2 hr at 60 mph, then drives 2 hr at 70 mph.  We expect his average speed to be 65 mph.

            S = ( (2)60 + 2(70)) / (2+2)

            S = (120+140)/4

             S =260 / 4

             S =  65

 

Now, realize that if the times for the two segments are unequal, we will be calculating a "weighted average."

 

So, try this problem:  Bill drives 2 hr at 60 mph, then 1 hr at 66 mph.  What's his average speed?

Answer 2

Distance = (Rate)(Time)

 

Average speed = (Total distance)/(Total time)

 

                     = [(60)(2) + 70H]/(2+H)

 

                     = (120 + 70H)/(H + 2)