# Problem of the Week Problem B and Solution Yukon Do It!

## Problem

The Yukon Quest is one of the most famous endurance sled dog races in the world. The total distance the race covers is $$1635\,$$km.

1. If the average distance travelled per day for each team is $$145\,$$km, how many days will it take, on average, for a team to complete the Yukon Quest?

2. A team of Alaskan huskies travels at $$15\,$$km per hour, with an $$18$$ minute rest after every three hours. A team of Siberian huskies runs more quickly at $$20\,$$km per hour, but requires a $$30$$ minute rest after every two hours.

On a certain day both teams travel $$145\,$$km. Create a broken-line graph of distance versus time for each team for that day.

Suggestion: You may find it helpful to first construct a table for each team, matching the total distance travelled with the elapsed time for each interval of travel and rest.

3. Suppose that the weight of some additional equipment slows the average speed of the Siberian huskies by $$5\,$$km per hour. If the team still travels $$145\,$$km in a day, by how many minutes will this increase their travel time for the day?

## Solution

1. Since the average (mean) distance traveled each day is $$145\,$$km, and the total distance is $$1635\,$$km, the number of days to complete the race is $$1635\div 145\approx 11.276$$ days. Thus, on average, the teams would finish on the $$12$$th day.

2. A broken-line graph of distance versus time for each team is shown below.

Here is the table for the Alaskan husky team.

Interval Type Start Time End Time Interval Distance Travelled (km) Total Distance Travelled (km)
Travel $$0$$ $$3$$ hrs $$45$$ $$45$$
Rest $$3$$ hrs $$3$$ hrs $$18$$ min $$0$$ $$45$$
Travel $$3$$ hrs $$18$$ min $$6$$ hrs $$18$$ min $$45$$ $$90$$
Rest $$6$$ hrs $$18$$ min $$6$$ hrs $$36$$ min $$0$$ $$90$$
Travel $$6$$ hrs $$36$$ min $$9$$ hrs $$36$$ min $$45$$ $$135$$
Rest $$9$$ hrs $$36$$ min $$9$$ hrs $$54$$ min $$0$$ $$135$$
Travel $$9$$ hrs $$54$$ min $$10$$ hrs $$34$$ min $$10$$ $$145$$

Here is the table for the Siberian husky team.

Interval Type Start Time End Time Interval Distance Travelled (km) Total Distance Travelled (km)
Travel $$0$$ $$2$$ hrs $$40$$ $$40$$
Rest $$2$$ hrs $$2$$ hrs $$30$$ min $$0$$ $$40$$
Travel $$2$$ hrs $$30$$ min $$4$$ hrs $$30$$ min $$40$$ $$80$$
Rest $$4$$ hrs $$30$$ min $$5$$ hrs $$0$$ $$80$$
Travel $$5$$ hrs $$7$$ hrs $$40$$ $$120$$
Rest $$7$$ hrs $$7$$ hrs $$30$$ min $$0$$ $$120$$
Travel $$7$$ hrs $$30$$ min $$8$$ hrs $$45$$ min $$25$$ $$145$$

Note that the Alaskan huskies travel $$135\,$$km in three segments of $$3$$ hours and $$18$$ minutes each, plus $$10\,$$km in a final segment of $$40$$ minutes. This gives a total time of $$10$$ hours and $$34$$ minutes. The Siberian huskies travel $$120\,$$km in three segments of $$2$$ hours and $$30$$ minutes each, plus $$25\,$$km in a final segment of $$1$$ hour and $$15$$ minutes. This gives a total time of $$8$$ hours and $$45$$ minutes.

3. Since the average speed of the Siberian huskies is now only $$15\,$$km per hour, they will travel only $$30\,$$km in each $$2$$ hour segment. Thus, they will now travel $$120\,$$km in four segments of $$2$$ hours and $$30$$ minutes each. That is, they will travel $$120\,$$km in $$10$$ hours. They will travel the last $$25\,$$km in a final segment of $$100$$ min, or $$1$$ hour and $$40$$ minutes. This gives a total time of $$11$$ hours and $$40$$ minutes. Thus, their time for the day has increased by $$2$$ hours and $$55$$ minutes, or $$175$$ minutes.