Question

An airplane is flying towards a radar station at a constant height of 6 km above the ground. If the distance s between the airplane and the radar station is decreasing at a rate of 400 km per hour when s=10 km., what is the horizontal speed of the plane? Make sure your answer includes units.

Answer 1

To find the horizontal speed of the airplane, the Pythagorean theorem is used. At the instance when the distance s is 10 km and the altitude is 6 km, with s decreasing at 400 km/h, the horizontal speed of the airplane is determined to be 400 km/h.

Step-by-step explanation:

The question asks to find the horizontal speed of an airplane flying at a constant altitude of 6 km and approaching a radar station. Given that the distance between the airplane and the radar station is decreasing at 400 km/h when the distance s is 10 km, we can use the Pythagorean theorem to solve for the horizontal component of the airplane's speed.

We have a right triangle where the altitude (6 km) is one leg, the distance s (10 km) is the hypotenuse, and the horizontal distance we need to find is the other leg. We can represent the horizontal speed as v. By the Pythagorean theorem:

v^2 + 6^2 = 10^2,
v^2 = 100 - 36,
v^2 = 64,
v = 8 km/h.

However, since the distance s is decreasing at 400 km/h, this rate is the hypotenuse of the right triangle formed every hour. Therefore, we use the relationship:

(400 km/h)^2 = (horizontal speed)^2 + (vertical speed)^2

The vertical component is zero as the plane is at a constant height. Solving for the horizontal speed:

(horizontal speed)^2 = (400 km/h)^2
horizontal speed = 400 km/h.

Therefore, the horizontal speed of the airplane is 400 km/h.

Answer 2

To solve this problem, we can use the concept of related rates. We are given that the distance s between the airplane and the radar station is decreasing at a rate of 400 km per hour when s = 10 km. We need to find the horizontal speed of the plane.

Let's denote the horizontal speed of the plane as v. Since the plane is flying at a constant height of 6 km above the ground, we can consider a right triangle formed by the airplane, the radar station, and the ground. The distance between the airplane and the radar station is the hypotenuse of this triangle, and the height of the triangle is 6 km.

Using the Pythagorean theorem, we have:

s^2 = (v^2) + (6^2)

Differentiating both sides of the equation with respect to time t, we get:

2s(ds/dt) = 2v(dv/dt)

Since ds/dt is the rate at which the distance s is changing (given as -400 km/h) and s = 10 km, we can substitute these values into the equation:

2(10)(-400) = 2v(dv/dt)

Simplifying further:

-8000 = 2v(dv/dt)

Now, we need to find the value of dv/dt, which represents the rate at which the horizontal speed of the plane is changing. Rearranging the equation, we have:

dv/dt = -8000 / (2v)

Given that s = 10 km, we can substitute this value into the equation for v:

10 = (v^2) + (6^2)

10 = v^2 + 36

v^2 = 10 - 36

v^2 = -26

Since we are dealing with speeds, we can discard the negative value. Therefore, the horizontal speed of the plane is v = √26 km/h (approximately 5.1 km/h).