Radian story problem help?

[zachstorm]

Abe and Bob are running along a circular track. They start out together, but run in opposite directions. Abe runs at 4 meters per second, and it takes him 92 seconds to complete each lap of the track. Bob takes 86 seconds to complete each lap.

a) What is Abe's angular speed in radians per second?
b) What is the radius of the track?
c) When do Abe and Bob pass each other for the first time?

 

[soroban]

Hello, zachstorm!

Abe and Bob are running along a circular track.
They start out together, but run in opposite directions.
Abe runs at 4 meters per second, and it takes him 92 seconds to complete each lap of the track.
Bob takes 86 seconds to complete each lap.

a) What is Abe's angular speed in radians per second?

\(\displaystyle \text{Abe runs }2\pi\text{ radians in 92 seconds.}\)

. . \(\displaystyle \text{His angular speed is: }\:\frac{2\pi}{92} \:\approx\: 0.0683\text{ rad/sec}\)

b) What is the radius of the track?

\(\displaystyle \text{Abe runs a lap in 92 seconds at 4 m/sec.}\)
\(\displaystyle \text{The circumference of the track is: }4 \times 92 \:=\:368\text{ m.}\)

\(\displaystyle \text{Since }C \,=\,2\pi r\text{, we have: }\:2\pi r \:=\:368 \quad\Rightarrow\quad r \:=\:\frac{184}{\pi} \:\approx\:58.57\text{ m.}\)

c) When do Abe and Bob pass each other for the first time?

\(\displaystyle \text{Abe runs at 4 m/s.}\)
. . \(\displaystyle \text{In }t\text{ seconds, he runs }4t\text{ meters.}\)
\(\displaystyle \text{Bob runs 368 m in 86 seconds. }\;\text{His speed is: }\frac{368}{86} \:=\:\frac{184}{43}\text{ m/sec.}\)
. . \(\displaystyle \text{In }t\text{ seconds, he runs }\frac{184}{43}t\text{ meters.}\)

\(\displaystyle \text{Together, in }t\text{ seconds, they run a full lap, 368 meters.}\)

. . \(\displaystyle 4t + \frac{184}{43}t \:=\:369 \quad\Rightarrow\quad \frac{356}{43}t \:=\:368 \quad\Rightarrow\quad t \:=\:\frac{3956}{89}\)

\(\displaystyle \text{They meet for the first time in about }44.45\text{ seconds.}\)