Surya is competing in a triathlon that includes swimming, bicycling and running. For the first two segments of the race, Surya swam at an average rate of 4 kilometers per hour and bicycled at an average rate of 40 kilometers per hour. It took 81.75 minutes for Surya to swim and cycle a combined total of 43.29 kilometers, completing the first two segments of the race. How long, in minutes, did it take him to complete the swimming portion of the race? Express your answer to the nearest hundredth.

If an athlete who completes this triathlon swims, bicycles and runs a combined total of 52.95 kilometers, the distance the athlete swam is what percent of the total race distance? Express your answer as a percent to the nearest hundredth.

At what average rate, in kilometers per hour, must Surya run the third, and final, portion of the race if he wishes to complete the entire race in no more than 140 minutes? Express your answer to the nearest hundredth.

1) 18.68 minutes

2) 2.34 %

3) 10.01 km/h

Step-by-step explanation:

1)

In the first part (swimming part), Surya's velocity is

where

is the length of the swimming part

is the time taken to complete the swimming part

We can rewrite this equation as

(1)

In the second part (biking part), Surya's velocity is

where

is the length of the biking part

is the time taken to complete the biking part

We can rewrithe this equation as

(2)

The total length of the two segments of the race is:

(3)

And the total time taken to complete them is

 (4)

Substituting (1) and (2) into (3),

(5)

From (4), we find:

And substituting into (5), we can find t1, the time taken to complete the swimming part of the race:

2)

In this part, we are told that the total distance covered by an athlete completing the swim + bycicle + run is

d = 52.95 km

This distance can be written as

where

is the length of the swimming part

is the length of the biking part

is the length of the running part

From the previous part, we can write the length of the swimming part as

where:

is the velocity in the swimming part

is the time taken for the swimming part

So we have

So, the percent of the swimming part over the total is:

3)

The total time for the entire race must be

And this total time can be written as

where

is the time taken to complete the swimming part

is the time taken to complete the biking part

is the time taken to complete the running part

From part 1) we know that

and

So the time to complete the running part must be

The distance of the last segment is

where is given by

So,

So the average velocity in the 3rd segment is