Having completed another stimulating school day, Michael and Alyssia both leave at the same time. They walk together for a while and then go their separate ways. These paths make a 60° angle with one another. Michael toddles along at 2ft/sec, while Alyssia speeds off at 3ft/sec. How far apart are they after 2 minutes?
after t seconds, Michael has gone 2t ft and Alyssa has gone 3t feet.
So, the distance z between them using the law of cosines, is found using
z^2 = (2t)^2 + (3t)^2 - 2(2t)(3t)cos60°
so, plug in your time (in seconds!) and find z.
z^2 = 240^2 + 360^2 - 2(240)(360)cos60°
z^2 = 100800
z = 317.5
Is this correct?
To find out how far apart Michael and Alyssia are after 2 minutes, we need to calculate the distances each of them has traveled in that time.
First, let's convert the 2 minutes to seconds, as their speeds are given in feet per second: 2 minutes * 60 seconds/minute = 120 seconds.
Since both Michael and Alyssia walk in a straight line, we can use basic trigonometry to find the distances they have traveled.
Let's consider Michael's path first. We can calculate the distance he has traveled using the formula distance = speed * time. Michael's speed is 2ft/sec, and the time is 120 seconds.
distance_michael = 2ft/sec * 120 seconds = 240ft
Now, let's consider Alyssia's path. Since the angle between their paths is 60°, the distance Alyssia has traveled can be found by calculating the side adjacent to the 60° angle in a right triangle.
We can use the formula distance = speed * time for Alyssia's path as well. Alyssia's speed is 3ft/sec, and the time is also 120 seconds.
distance_alyssia = 3ft/sec * 120 seconds = 360ft
Now, to find how far apart they are after 2 minutes, we need to use the Law of Cosines.
The Law of Cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice the product of those two sides multiplied by the cosine of the included angle.
In this case, we want to find the distance between Michael and Alyssia after 2 minutes, which is the side opposite to the 60° angle. So, we can set up the equation:
distance^2 = distance_michael^2 + distance_alyssia^2 - 2 * distance_michael * distance_alyssia * cos(60°)
Let's plug in the values we calculated:
distance^2 = 240^2 + 360^2 - 2 * 240 * 360 * cos(60°)
Using a calculator to evaluate this equation, we find:
distance^2 = 57600 + 129600 - 172800 * 0.5 = 57600 + 129600 - 86400 = 120800
Taking the square root of both sides to find the distance:
distance = sqrt(120800) ≈ 347.88ft
Therefore, after 2 minutes, Michael and Alyssia are approximately 347.88 feet apart.