Questions: A team of astronauts is training for being weightless in space by riding in an airplane which is flying in a sinusoidal curve - the airplane flies high and then turns steeply down to mimic free fall, and hence weightlessness for the astronauts, before flying back to the top of the curve. The path of the airplane can be modeled using the function h(x)= 10000 cos(0.0001 x)+20000 in which h indicates the height in feet. What is the minimum height of the airplane? 5,000 ft 10,000 ft 15,000 ft 20,000 ft 30,00 ft

A team of astronauts is training for being weightless in space by riding in an airplane which is flying in a sinusoidal curve - the airplane flies high and then turns steeply down to mimic free fall, and hence weightlessness for the astronauts, before flying back to the top of the curve. The path of the airplane can be modeled using the function h(x)= 10000 cos(0.0001 x)+20000 in which h indicates the height in feet. What is the minimum height of the airplane?
5,000 ft
10,000 ft
15,000 ft
20,000 ft
30,00 ft

Transcript text: A team of astronauts is training for being weightless in space by riding in an airplane which is flying in a sinusoidal curve - the airplane flies high and then turns steeply down to mimic free fall, and hence weightlessness for the astronauts, before flying back to the top of the curve. The path of the airplane can be modeled using the function $h(x)=$ $10000 \cos (0.0001 x)+20000$ in which $h$ indicates the height in feet. What is the minimum height of the airplane? $5,000 \mathrm{ft}$ $10,000 \mathrm{ft}$ $15,000 \mathrm{ft}$ $20,000 \mathrm{ft}$ $30,00 \mathrm{ft}$

Solution

Solution Steps

Step 1: Understand the Function

The function given is $ h(x) = 10000 \cos (0.0001 x) + 20000 $. This is a cosine function that models the height of the airplane. The general form of a cosine function is $ y = A \cos(Bx) + C $, where:

$ A $ is the amplitude,
$ B $ affects the period of the function,
$ C $ is the vertical shift.

Step 2: Identify the Amplitude and Vertical Shift

In the function $ h(x) = 10000 \cos (0.0001 x) + 20000 $:

The amplitude $ A = 10000 $. This means the maximum deviation from the midline (vertical shift) is 10,000 feet.
The vertical shift $ C = 20000 $. This means the midline of the cosine wave is at 20,000 feet.

Step 3: Calculate the Minimum Height

The minimum value of the cosine function $\cos(\theta)$ is $-1$. Therefore, the minimum height of the airplane can be calculated as follows:

\[ h_{\text{min}} = 10000 \times (-1) + 20000 = -10000 + 20000 = 10000 \text{ feet} \]