2440.12 – Orthodontist

Phoebe Beebe has an appointment with her orthodontist Dr. Jane Pullem (and nurse Willy Fillem) at 2:55 PM this afternoon. Her brother Led will drive her there, and they will leave right after school. If they go 60 kph (about 35 mph) the whole way, they will arrive 5 minutes early. But if they drive 40 kph (about 25 mph) the whole way, they will arrive 5 minutes late. What time does school get out, and how far do they have to drive?

Solution

Consider the chart shown below. We'll use d=r⋅td = r \cdot t, where dd is the distance to the orthodontist's office in kilometers,

and tt is the time in hours that it takes to drive the distance at 6060 kph. Since d=dd = d, we have

60t=40(t+1/6)60t=40t+40/6360t=240t+40120t=40t=40/120=1/3 hour.\begin{aligned}60t &= 40(t + 1/6) \\60t &= 40t + 40/6 \\360t &= 240t + 40 \\120t &= 40 \\t &= 40/120 = 1/3 \text{ hour.}\end{aligned}

At 6060 kph they drive for 2020 minutes and arrive 55 minutes early, at 2:50. So they left at 2:30, the time that school got out. The distance is 60⋅13=2060 \cdot \dfrac{1}{3} = 20 km.