Question

A plane flying with a constant speed of 360 km/h passes over a ground radar station at an altitude of 1 km and climbs at an angle of 30°. At what rate (in km/h) is the distance from the plane to the radar station increasing a minute later? (Round your answer to the nearest whole number.)

Answer 1

Answer: stated down below

Step-by-step explanation:

To solve this problem, we can use trigonometry and related rates.

Let's define the following variables:

h: altitude of the plane from the ground radar station (in km)

d: horizontal distance from the plane to the radar station (in km)

Given:

The plane is flying at a constant speed of 360 km/h.

The altitude of the plane is 1 km.

The plane climbs at an angle of 30°.

We want to find the rate at which the distance from the plane to the radar station is increasing a minute later.

First, let's consider the right triangle formed by the plane, the radar station, and the distance the plane has traveled in one minute (which is equal to its speed, 360 km/h).

In this right triangle, the angle between the hypotenuse (the distance the plane has traveled in one minute) and the base (horizontal distance d) is 30°. The opposite side of this angle is the altitude of the plane, h.

Using trigonometry, we can write the following equation:

tan(30°) = h / d

tan(30°) = (1 km) / d

√3/3 = 1 / d

d = 3 / √3 = √3 km

Now, to find the rate at which the distance is increasing, we need to differentiate the equation with respect to time.

Differentiating both sides with respect to time:

d/dt (d) = d/dt (√3 km)

The derivative of d with respect to time represents the rate at which the distance is changing, which is what we want to find.

Since the speed of the plane is constant at 360 km/h, the derivative of d with respect to time is equal to the speed:

d/dt (d) = 360 km/h

Therefore, the rate at which the distance from the plane to the radar station is increasing a minute later is approximately 360 km/h (rounded to the nearest whole number).

Note: The rate of increase of the distance is independent of the altitude of the plane.

Answer 2

The distance from the plane to the radar station is increasing at a rate of 34 km/h (rounded to the nearest whole number).

Step-by-step explanation:

Convert units: Convert the altitude and speed to meters per second to avoid unnecessary conversions later:

Altitude: 1 km * 1000 m/km = 1000 m

Speed: 360 km/h * (1h/3600s) * (1000 m/km) = 100 m/s

Calculate horizontal distance: We need the horizontal distance between the plane and the radar station to determine the rate of change. This can be found using trigonometry:

Horizontal distance (h) = Altitude (a) / tan(Climbing angle)

h = 1000 m / tan(30°) ≈ 1732 m

Calculate vertical distance change: After one minute, the plane travels vertically:

Vertical distance (v) = Speed * Time

v = 100 m/s * (60s/min) = 6000 m

Apply Pythagorean theorem: The total distance (d) between the plane and the radar station is the hypotenuse of a right triangle with legs h and v:

d^2 = h^2 + v^2

d^2 ≈ 1732^2 + 6000^2

d ≈ 6204 m

Calculate rate of change: We need the rate of change of the hypotenuse (d) with respect to time (t). This can be approximated using the difference quotient in one minute:

Rate of change ≈ (d(t + 1 min) - d(t)) / (1 min)

Assume small time step: Since the time step is small (1 minute), we can use the same hypotenuse value (d) for both time points without significant error. This simplifies the calculation:

Rate of change ≈ (6204 m - 6204 m) / (1 min)

Rate of change ≈ 0 m/min (negligible change in one minute)

Horizontal rate of change: The actual rate of change is dominated by the horizontal movement. As the angle is 30°, the horizontal component of the speed (u) is:

u = Speed * cos(Climbing angle)

u = 100 m/s * cos(30°) ≈ 86.6 m/s

Convert to km/h:

Rate of change ≈ u ≈ 86.6 m/s * (1h/3600s) * (1000 m/km) ≈ 246 km/h

Therefore, although the overall distance change in one minute is negligible, the horizontal movement contributes to a significant rate of change of approximately 246 km/h. Rounding to the nearest whole number, the rate of distance increase is 34 km/h.