.Two hikers begin at the same location and walk in perpendicular directions. Hiker A travels due north at a rate of 3 miles per hour; Hiker B travels due east at a rate of 5 miles per hour. At what rate is the distance between the two hikers changing 4 hours into the hike?
let the time passed be t hours
Simple right-angled triangles
let the distance between them be D
D^2 = (3t)^2 + (5t)^2
D^2 = 34t^2
D = √34 t
dD/dt = √34
So the distance between them is changing at a constant rate of √34 mph, independent of the time
To find the rate at which the distance between the two hikers is changing, we can use the concept of relative velocity. The distance between the two hikers can be considered as the hypotenuse of a right triangle, with one side representing the distance traveled by Hiker A (3 miles/hour for 4 hours) and the other side representing the distance traveled by Hiker B (5 miles/hour for 4 hours).
The distance traveled by Hiker A in 4 hours is 3 miles/hour * 4 hours = 12 miles.
The distance traveled by Hiker B in 4 hours is 5 miles/hour * 4 hours = 20 miles.
Now, we can apply the Pythagorean theorem to find the distance between the two hikers after 4 hours. The distance is given by:
Distance = √(12^2 + 20^2) = √(144 + 400) = √544 = 23.32 miles. (Approximately)
To find the rate at which the distance is changing after 4 hours, we need to differentiate this distance equation with respect to time. However, this is not explicitly mentioned in the question. So, we can assume that both hikers are moving at a constant speed and that the distance between them is changing at a constant rate.
Therefore, the rate at which the distance between the two hikers is changing 4 hours into the hike is 0 miles/hour.
To find the rate at which the distance between the two hikers is changing, we can use the Pythagorean theorem to determine the distance between them at any given time.
Let's denote the distance traveled by hiker A as dA and the distance traveled by hiker B as dB.
Since Hiker A is traveling due north at a rate of 3 miles per hour, after 4 hours, Hiker A would have covered a distance of dA = 3 mph * 4 hours = 12 miles.
Similarly, Hiker B is traveling due east at a rate of 5 miles per hour, so after 4 hours, Hiker B would have covered a distance of dB = 5 mph * 4 hours = 20 miles.
Using the Pythagorean theorem, the distance between the two hikers is given by:
distance = sqrt(dA^2 + dB^2)
Plugging in the values we found:
distance = sqrt(12^2 + 20^2)
= sqrt(144 + 400)
= sqrt(544)
= 23.32 miles (approximately)
To find the rate at which the distance is changing, we can differentiate the equation for distance with respect to time. However, since we're given a specific time, we can simply calculate the rate of change using the distance covered by each hiker in that specific time period.
After 4 hours, both hikers have covered the full distances, so the rate at which the distance between them is changing at that time is 0 miles per hour.