(4 pts) Imagine that there is a circular track given by the equation x2+y2=2102500, where both x and y are measured in units of feet. At time t=0, Robin is at point (1450,0) and Shannon is at point (−170,1440). Both are running counterclockwise, with Shannon running at 2.3 feet per second and Robin running at 3.6 feet per second.
Robin runs until he catches Shannon. How long does that take (in seconds)?
At what point do Robin and Shannon meet?
radius of circle = 1450 ft
circumference of whole track = 2π(1450)
= 2900π ft
angle of sector between them
= 90 + arctan(1440/170) = 173.267°
Distance Robin has to make up is
173.267/360 = x/2900π
x = 4384.9155 ft
He is gaining on him at a rate of
1.3 ft/sec
so to catch up on the 4384.9155 ft would take
4384.9155/1.3 seconds
= 3373.01 seconds
unreasonable answer?
no, unreasonable question.
To find out when Robin catches Shannon, we need to find the point where their paths intersect on the circular track.
Given the equation of the circular track: x^2 + y^2 = 2102500, we can substitute the x-coordinate and y-coordinate of the two runners to find the time it takes for Robin to catch Shannon.
1. Robin's position at t=0: (1450, 0)
2. Shannon's position at t=0: (-170, 1440)
3. Let's assume the time Robin takes to catch Shannon is t seconds.
To determine the equation for Robin's position, we need to consider his speed and the distance he covers:
4. Robin's position at time t: (1450 + 3.6t, 0)
To find the equation for Shannon's position, we need to consider her speed and the distance she covers:
5. Shannon's position at time t: (-170 + 2.3t, 1440)
Now, we need to find the values of t where Robin and Shannon's positions intersect.
Substituting equations (4) and (5) into the equation of the circular track, we get:
(1450 + 3.6t)^2 + (2.3t - 1440)^2 = 2102500
Expanding and simplifying:
5.8t^2 + 15060t - 2174400 = 0
Now, we can solve this quadratic equation to find the values of t.
Using the quadratic formula, t = (-b ± sqrt(b^2 - 4ac))/(2a), where:
a = 5.8
b = 15060
c = -2174400
By substituting the values into the quadratic formula, we get:
t = (-15060 ± sqrt(15060^2 - 4 * 5.8 * -2174400))/(2 * 5.8)
Simplifying the equation, we have:
t = (-15060 ± sqrt(226195600 - (-50365760)))/(11.6)
t = (-15060 ± sqrt(276561360))/(11.6)
After calculating the values:
t ≈ 110.4 seconds or t ≈ -29.4 seconds
Since time cannot be negative, the final answer is:
Robin catches Shannon after approximately 110.4 seconds.
To find the point where Robin and Shannon meet, we can substitute the value of t into any of their equations:
Using Robin's position at time t:
x = 1450 + 3.6(110.4) ≈ 1862.4
y = 0
Therefore, Robin and Shannon meet approximately at the point (1862.4, 0).
To find the time it takes for Robin to catch Shannon, we need to calculate the time it takes for Robin to reach the same position as Shannon on the circular track.
First, let's find the angle that represents Shannon's current position on the track. We can use trigonometry to calculate this angle:
θ = arctan(y/x)
For Shannon's coordinates at t=0, we have x = -170 and y = 1440. Plugging these values into the equation, we get:
θ = arctan(1440/(-170))
Using a calculator, we find that the angle is approximately 280.91 degrees.
Now, let's determine Robin's starting angle. Since Robin starts at point (1450, 0), his angle is 0 degrees.
To find the distance that Robin needs to cover to catch Shannon, we calculate the difference between their angles on the circular track:
Δθ = |θ_Shannon - θ_Robin|
Δθ = |280.91 - 0|
Δθ = 280.91 degrees
Next, we need to convert the distance Robin covers to the time it takes, using their respective speeds.
Let's define the speed ratio as:
Speed ratio = Robin's speed / Shannon's speed
Speed ratio = 3.6 / 2.3
Speed ratio ≈ 1.5652
Now, we can calculate the time it takes for Robin to catch Shannon by dividing the angular distance by the speed ratio:
Time = Δθ / Speed ratio
Time = 280.91 / 1.5652
Time ≈ 179.57 seconds
So, it takes approximately 179.57 seconds for Robin to catch Shannon on the circular track.
To find the point where Robin and Shannon meet, we need to calculate the coordinates of the point on the circular track corresponding to the time it took for Robin to catch Shannon.
We know that Shannon's coordinates at t=0 are (-170, 1440), and Robin's start position is (1450, 0).
Using their respective speeds, we can find the positions of Shannon and Robin at the meeting point:
Shannon's x-coordinate = -170 + (2.3 * Time)
Shannon's y-coordinate = 1440
Robin's x-coordinate = 1450 + (3.6 * Time)
Robin's y-coordinate = 0
By substituting the calculated time into these equations, we can find the coordinates where Robin and Shannon meet.