A helicopter is 400 miles directly north of Ghana and is flying at 20 miles per hour. A second helicopter is 300 miles east of Ghana and is flying west at 15 mph. What is the rate of change of the distance between the helicopters?
You don't say which way the first helicopter is flying, north or south or ...
I will assume it is flying south.
Make a right-angled triangle with y as the remaining distance to Ghana and x as the remaining distance.
let d be the distance between them
d^2 = y^2 + x^2
2d dd/dt = 2ydy/dt + 2xdx/dt
when x=300 and y = 400
d^2 = 300^2 + 400^2
d = 500 (did you recognize the 3,4, 5 triangle ?)
dd/dt = (400(-20) + 300(-15))/(2(500)) = 12.5
To find the rate of change of the distance between the helicopters, we need to determine how the distance is changing with respect to time.
Let's break down the problem step by step:
1. Assume that the helicopters start moving at the same time.
2. Let's use the Pythagorean theorem to find the distance between the two helicopters at a given time. The theorem states that the square of the hypotenuse (the distance between the helicopters) is equal to the sum of the squares of the other two sides.
So, the distance between the helicopters can be calculated using the formula:
distance^2 = (distance north)^2 + (distance east)^2
3. Initially, the helicopters are 400 miles north and 300 miles east of Ghana, respectively. Therefore, their initial distance (D) is given by:
D = √((400)^2 + (300)^2)
4. Now, let's consider how the distances north and east change over time.
The helicopter north of Ghana is flying at a speed of 20 mph, so its distance north is changing at a rate of 20 miles per hour.
Likewise, the helicopter east of Ghana is flying west (which is in the negative direction) at a speed of 15 mph, so its distance east is changing at a rate of -15 miles per hour.
5. Since the helicopters are moving independently of each other, we treat their distances north and east as separate variables.
Let's denote the distance north as x and the distance east as y.
6. Given that x = 400 miles and y = 300 miles, the mathematical representation of the distance between the helicopters (D) is:
D = √(x^2 + y^2)
7. To find the rate of change of the distance between the helicopters (dD/dt), we can differentiate the equation D = √(x^2 + y^2) with respect to time (t):
dD/dt = (2x)(dx/dt) + (2y)(dy/dt)
Substituting the given rates of change (dx/dt = 20 mph, dy/dt = -15 mph), we can calculate the rate of change of the distance between the helicopters (dD/dt).