Final answer:
The question is asking for the rate at which the distance between a person and an airplane is increasing as the airplane flies away horizontally at a constant elevation. By utilizing the Pythagorean theorem and applying derivatives we can find the answer. Once we calculate the slope of this triangle, we can use it to calculate the rate at which the distance between the person and the plane is increasing as it flies overhead.
Step-by-step explanation:
To find at what rate the distance between the person and the plane is increasing we can use the Pythagorean theorem to represent the scenario and differentiate to find the rate of change of distance with respect to time.
Let's designate the distance between the person and the base of the radio tower as x, the constant elevation of the plane as y = 4040 ft, and the distance between the person and the plane as z. The plane is flying horizontally away from the person at 600 ft/s, which tells us that x is also changing with respect to time, which we can denote as dx/dt.
The relationship between x, y, and z can be defined as:
z² = x² + y²
By differentiating both sides of this equation with respect to t (time), we obtain:
2z(dz/dt) = 2x(dx/dt) (since y is constant and its derivative is zero)
Next, we solve for dz/dt which represents the rate at which the distance from the person to the plane is increasing. Substituting in the known values:
x = 3055 ft (the initial distance from the person to the tower/base of the plane)
dx/dt = 600 ft/s (the speed of the plane)
y = 4040 ft (the elevation of the plane, which remains constant)
We find that:
z = √(x² + y²) = √(3055² + 4040²)
Calculating the value of z, and then substituting x, dx/dt, and z into the differentiated equation, we will obtain the value of dz/dt, which is the rate we seek.