Question

Suppose you have entered 79​-mile biathlon that consists of a run and a bicycle race. During your​ run, your average velocity is miles 5per​ hour, and during your bicycle​ race, your average velocity is 23 miles per hour. You finish the race in 5 hours. What is the distance of the​ run? What is the distance of the bicycle​ race?

Answer 1

• run: 10 miles
• bike race: 69 miles

Explanation:

Let r represent the distance run. Then 79-r is the distance biked. The total time is ...

time = distance/speed

5 = (r/5) +(79-r)/23 . . . . sum of time running and time biking

575 = 23r +5(79 -r) . . . . multiply by 115

180 = 18r . . . . . . . . . . . . . subtract 395

10 = r . . . . . . . . . . . divide by 18

79-r = 69 . . . . . . distance biking

The distance of the run is 10 miles; the bicycle race is 69 miles.

Answer 2

The distance of the run is 10 miles

The distance of the bicycle race is 69 miles

Explanation:

Let:

• r = the distance for the run
• 79 - r = the distance for the bicycle race
• t = the time of the run
• 5 - t = the time of the bicycle race

The run: r = 5t

The bicycle race: 79 - r = 23(5 - t)

Simplify the bicycle race:

• 79 - r = 115 - 23t
• -r = 36 - 23t
• r = 23t - 36

Now we have two equations set up for r

• r = 5t
• r = 23t - 36

We know that r = r, so 5t must equal 23t - 36

• 5t = 23t - 36
• 5t + 36 = 23t
• 36 = 18t
• t = 2

Now we know that the run took 2 hours.

Plug it back into the equation for the run to find the distance for the run

• r = 5t
• r = 5(2)
• r = 10

The run is 10 miles

To solve for the bicycle race, we simply subtract 10 from 79, because we know that the two distances combined are 79.

• 79 - 10 = 69

The bicycle race is 69 miles.

-Chetan K