Question

1. a plane is going to fly 200 miles at a planned speed of 530 miles per hour. the flight will have an average headwind of w miles per hour the entire time, meaning the plane is flying directly against the wind. the time t, in hours of the flight is a function of the speed of the headwind w, in miles per hour, and can be modeled by

t(w)=\frac{200}{530 - w}

here is the graph of y = t(w):

a. what does t(150) mean in this situation?

b. at what value of w does the graph have a vertical asymptote? explain how you know and what this asymptote means in the situation.

Explanation:

a.

In the function \( T(w)=\frac{200}{530 - w} \), the input \( w \) represents the head - wind speed. When we evaluate \( T(150) \), we substitute \( w = 150 \) into the function. The function \( T(w) \) gives the time of the flight. So \( T(150) \) means the time (in hours) of the flight when the head - wind speed is \( 150 \) miles per hour.

For a rational function \( y=\frac{f(x)}{g(x)} \), vertical asymptotes occur where \( g(x)=0 \). In the function \( T(w)=\frac{200}{530 - w} \), we set the denominator equal to zero: \( 530 - w=0 \). Solving for \( w \), we get \( w = 530 \).

In the context of the problem, the plane's speed relative to the ground is \( 530−w \). When \( w = 530 \), the plane's speed relative to the ground is \( 530-530=0 \) miles per hour. This means the plane is not moving forward (it is stationary relative to the ground), and the time of the flight \( T(w) \) approaches infinity.

Answer:

\( T(150) \) represents the time (in hours) of the flight when the average head - wind speed is \( 150 \) miles per hour.

b.