Beth leaves her home on her bike, traveling down a long country road, at
the rate of 8 mph. her brother, brandon, left 15 minutes after beth on his
golf cart along the same country road, traveling 12 mph. how long will it
take brandon to catch up with beth? it says the answer is 30 min how is that true?

In this question, you are given Beth velocity(8mph) and Brandon velocity(12mph). To answer this question, you will need to find the initial distance of Beth and Brandon. The calculation would be:

Initial distance= Beth velocity * duration= 8 mph * 15 minutes * hour/60 minutes= 2miles

After that, the calculation would be:

beth_velocity * time + initial distance = brandon_velocity * time
8mph * time + 2 miles= 12 mph* time
4mph * time = 2 miles
time= 2 miles / 4 mph= 1/2 hour = 30 minutes