Question

a plane flying horizontally at an altitude of 1 mile and a speed of 580 mi/h passes directly over a radar station. find the rate at which the distance from the plane to the station is increasing when it has a total distance of 5 miles away from the station. (round your answer to the nearest whole number.)

Answer 1

To find the rate at which the distance from the plane to the radar station is increasing, set up a proportion and differentiate both sides of the equation with respect to time. Substitute the given values and solve for the rate of increase.

Step-by-step explanation:

To find the rate at which the distance from the plane to the radar station is increasing, we can use the concept of similar triangles. Let's denote the distance between the radar station and the plane as x and the total distance of 5 miles as y. Since the plane is flying horizontally, the height of the triangle formed by the radar station, the plane, and the distance between them remains constant at 1 mile.

We can set up a proportion: x/1 = y/5, where x is the variable we want to solve for. Cross-multiplying, we get x = y/5. Differentiating both sides of the equation with respect to time, we get dx/dt = (dy/dt)/5.

Since the problem states that the speed of the plane is 580 mi/h, we can substitute dy/dt = 580 mi/h into the equation. Therefore, dx/dt = (580 mi/h)/5 = 116 mi/h. Rounding to the nearest whole number, the rate at which the distance from the plane to the radar station is increasing when it is 5 miles away is approximately 116 mi/h.

Answer 2

To find the rate at which the distance from the plane to the station is increasing, we can use the concept of similar triangles. By setting up a proportion and differentiating with respect to time, we can find that the rate is approximately 2320 mi/h.

Step-by-step explanation:

To find the rate at which the distance from the plane to the station is increasing, we can use the concept of similar triangles.

First, let's draw a diagram to represent the situation:

Diagram:

Let x be the distance from the plane to the station.

Using the concept of similar triangles, we can set up the following proportion:

(1 mile)/(x) = (5 miles)/(x + y)

Where y is the distance the plane has traveled away from the station.

Now, let's solve the proportion for y:

1(x + y) = 5x

x + y = 5x

y = 4x

Now, differentiate both sides of the equation with respect to time:

dy/dt = 4(dx/dt)

Since we are given dx/dt = 580 mi/h, we can substitute this into the equation:

dy/dt = 4(580 mi/h)

dy/dt ≈ 2320 mi/h

Therefore, the rate at which the distance from the plane to the station is increasing when it is 5 miles away from the station is approximately 2320 mi/h.