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.