Two people start from the same point. One walks east at 3 mi/h and the other walks northeast at 2 mi/h. How fast is the distance between the people changing after 15 minutes?
To find how fast the distance between the two people is changing, we can use the concept of derivatives. Let's consider the positions of the two people at any given point in time.
Let's assume the initial point of both individuals is (0,0) on a coordinate system. The first person walks east at a constant velocity of 3 miles per hour, so the position of that person after time t can be represented as (3t, 0).
The second person walks northeast, which is at a 45-degree angle to the east. This person covers a distance of 2 miles per hour. Since the angle is 45 degrees, the motion can be decomposed into the east and north components. The east component can be calculated as (2 cos 45)t = (2√2/2) t = √2t, and the north component is (2 sin 45)t = (2√2/2) t = √2t. Therefore, the position of the second person after time t can be represented as (√2t, √2t).
To find the distance between these two positions, we can use the distance formula:
Distance = √((Δx)^2 + (Δy)^2),
where Δx and Δy are the differences in the x and y coordinates, respectively.
So, the distance between the two people after time t is:
Distance = √((3t - √2t)^2 + (0 - √2t)^2)
= √((9t^2 - 6√2t^2 + 2t) + (2t))
= √(9t^2 - 6√2t^2 + 2t + 2t)
= √(9t^2 - 6√2t^2 + 4t)
Now, we can find how fast the distance is changing (dDistance/dt) after 15 minutes (t = 15 minutes = 0.25 hours).
Differentiating the distance equation with respect to time (t) gives:
dDistance/dt = (1/2)(9t^2 - 6√2t^2 + 4t)^(-1/2) * (18t - 6√2t + 4)
Substituting t = 0.25 into the above equation will give us the rate at which the distance between the two people is changing after 15 minutes.
dDistance/dt = (1/2)(9(0.25)^2 - 6√2(0.25)^2 + 4(0.25))^(-1/2) * (18(0.25) - 6√2(0.25) + 4)
= (1/2)(0.5625 - 0.375√2 + 1) ^ (-1/2) * (4.5 - 1.5√2 + 4)
= (1/2)(3.5625 - 0.375√2) ^ (-1/2) * (8.5 - 1.5√2)
So, after performing the calculations, we can determine how fast the distance between the two people is changing after 15 minutes.