Derivatives - chain rule

An airplane, flying horizontally at an altitude of 1 mile, passes directly over an observer. If the constant speed of the airplane is 400 miles per hour, how fast is its distance from the observer increasing 45 seconds later? Hint: note that in 45 seconds (3/4 * 1/60 = 1/80), the airplane goes 5 miles.

To find how fast the distance from the observer is increasing, we need to find the derivative of the distance function with respect to time.

Let's say the airplane is at a distance x miles from the observer at time t seconds.

From the given information, we know that the airplane is flying horizontally at a constant speed of 400 miles per hour. This means the rate of change of x with respect to t is given by:

dx/dt = 400

We also know that in 45 seconds, the airplane travels 5 miles. This gives us the relationship:

dx/dt = 5/(45 seconds)

Now, we need to find dz/dt, where z represents the distance between the observer and the airplane.

Using the Pythagorean theorem, we have:

z^2 = x^2 + 1^2

Differentiating both sides with respect to t using the chain rule, we get:

2z * dz/dt = 2x * dx/dt

Since we want to find dz/dt, we can rearrange the equation as follows:

dz/dt = (x * dx/dt) / z

Substituting the values we know, we have:

dz/dt = (x * 5/(45 seconds)) / z

Now, we need to find x when t = 45 seconds. Given that x = 5 miles in 45 seconds, we can substitute x = 5 into the equation:

dz/dt = (5 * 5/(45 seconds)) / z

Simplifying, we get:

dz/dt = (25/45) / z

To find the value of z, we can use the Pythagorean theorem again:

z = sqrt(x^2 + 1)
z = sqrt(5^2 + 1)
z = sqrt(26)

Substituting this value into the equation, we have:

dz/dt = (25/45) / sqrt(26)

Now we can calculate the numerical value of dz/dt using a calculator.