Boris and Stella are running on a circular track which has a
radius of 300 meters. They start running at the same time.
Boris starts at the northernmost point of the track, and
Stella starts at the easternmost point. Boris runs clockwise.
Stella runs counterclockwise, and she runs twice as fast as
Boris runs. They pass each other for the first time after 85
seconds. How far apart are Boris and Stella when they
have been running for 800 seconds?
They are 1/4 of a circle apart to begin.
Boris runs 1/12 of a circle and Stella runs 1/6 of a circle in 85 seconds.
So, Boris's angular speed is ω = 2π/85 and Stella's speed is 4π/85
So, at time t,
Boris's location is (300 sin 2π/85 * 800, 300 cos 2π/85 * 300)
Stella's location is (300 cos 4π/85 * 800, 300 sin 4π/85 * 300)
So just use the distance formula to get the answer.
To find out how far apart Boris and Stella are when they have been running for 800 seconds, we need to determine how many times they pass each other in that time and calculate the distance between them at each passing.
Since Boris and Stella start at opposite points on the circular track and run in opposite directions, they will pass each other every time they complete a full lap around the track. Let's calculate how long it takes for both Boris and Stella to complete one lap each.
The circumference of a circle is given by the formula C = 2 * π * r, where r is the radius of the circle. In this case, the radius is 300 meters.
C = 2 * π * 300
C ≈ 1884.96 meters
Therefore, it takes each of them approximately 1884.96 meters to complete one lap around the circular track.
Since Stella runs twice as fast as Boris, her speed is two times Boris's speed. Let's assume Boris's speed is S meters per second. Therefore, Stella's speed is 2S meters per second.
We know that Boris and Stella pass each other for the first time after 85 seconds. This means that they have covered a combined distance of 1884.96 meters in 85 seconds.
The distance Boris has covered in 85 seconds is S * 85 meters.
The distance Stella has covered in 85 seconds is 2S * 85 meters.
Their combined distance is S * 85 + 2S * 85 meters.
Now, we can calculate the speed of Boris:
Distance = Speed * Time
S * 85 = S * T, where T is the time it takes for Boris to cover one lap.
From this equation, we can solve for T:
T = (S * 85) / S
T = 85 seconds
Since Boris takes 85 seconds to cover one lap, Stella also takes 85 seconds to cover one lap because Stella runs twice as fast as Boris.
Now, let's find out how many times they pass each other in 800 seconds:
Number of times they pass = 800 / (85 + 85)
Number of times they pass = 800 / 170
Number of times they pass = 4.706
Since they cannot pass each other in fractions, we can round this down to the nearest whole number. Therefore, they pass each other a total of 4 times in 800 seconds.
Now, let's calculate the remaining distance between Boris and Stella after the 4th passing:
The distance between Boris and Stella after 4 laps = 1884.96 * (4 - 1) meters.
Since they've already passed each other 4 times, the remaining distance is the difference between the total distance and the distance covered in the 4 laps, which is:
Remaining distance = 1884.96 * (4 - 1) meters - (S * 85 + 2S * 85 meters) * 4
Now, we need to substitute the values into the formula and calculate the distance:
Remaining distance = 1884.96 * (4 - 1) - (S * 85 + 2S * 85) * 4
To solve this equation, we need to know the value of S, which is Boris's speed. Unfortunately, the problem does not provide this information.