Jacob can run a lap around a track in 3 minutes, and Betty can run a lap around the track in 5 minutes. If they both start at the exact same time, how long will it take until they cross the start line again together?

15 minutes until they cross the start line again together

Step-by-step explanation:

Jacob will finish his laps in these following minutes, supposing he starts at 0:

3 minutes, 6 minutes, 9 minutes, 12 minutes, 15 minutes,...

That is, all instants of time which are multiples of three.

Betty

5 minutes, 10 minutes, 15 minutes, 20 minutes, 25 minutes, ...

That is, all instants of time which are multiples of five.

If they both start at the exact same time, how long will it take until they cross the start line again together?

This is the lesser common multiple of 3 and 5, which is 15.

So 15 minutes until they cross the start line again together

step-by-step explanation:

x^2 -12x + 36 + y^2 = 36

x^2 + y^2 = 12x

r^2 = 12rcos(theta)

r = 12 cos(theta)