Question

i can find the solutions to rational equations and ignore extraneous solutions.

1. a plane is going to fly 300 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{300}{530 - w} \\)
here is the graph of \\( y = t(w) \\):
graph of t(w) with t (travel time) on y-axis and w (headwind speed) on x-axis, showing a curve that increases, with x from 0 to 600 and y from 0 to 10

a. what does \\( t(100) \\) 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:

Part a

Step1: Understand the function

The function \( T(w)=\frac{300}{530 - w} \) models the time \( T \) (in hours) of the flight as a function of the headwind speed \( w \) (in miles per hour). Here, \( w = 100 \) means the headwind speed is 100 miles per hour.

Step2: Substitute \( w = 100 \)

Substitute \( w=100 \) into the function \( T(w) \): \( T(100)=\frac{300}{530 - 100}=\frac{300}{430}\approx0.698 \) (or keep it as a fraction \( \frac{30}{43} \)). In the context, \( T(100) \) represents the time (in hours) the flight takes when the headwind speed is 100 miles per hour.

Step1: Find vertical asymptote of rational function

For a rational function \( y=\frac{f(w)}{g(w)} \), the vertical asymptote occurs where \( g(w) = 0 \) (and \( f(w)
eq0 \) at that point). For \( T(w)=\frac{300}{530 - w} \), set the denominator equal to zero: \( 530 - w=0 \).

Step2: Solve for \( w \)

Solving \( 530 - w = 0 \) gives \( w = 530 \).

Step3: Interpret the asymptote

In the context, the denominator \( 530 - w \) represents the effective speed of the plane (since it's flying against the wind, effective speed = planned speed - headwind speed). When \( w = 530 \), the effective speed becomes \( 530 - 530=0 \) miles per hour. This means the plane's effective speed is zero, so it would never complete the 300 - mile flight (time would approach infinity as \( w \) approaches 530 from the left, since \( w \) can't be more than 530 in a real - world sense here because if \( w>530 \), the denominator would be negative and the time would be negative which doesn't make sense for time in this context). So the vertical asymptote at \( w = 530 \) means that as the headwind speed approaches 530 miles per hour (from speeds less than 530), the time taken for the flight approaches infinity, and a headwind speed of 530 miles per hour would mean the plane has no forward motion relative to the ground, so it can't complete the flight.

Answer:

\( T(100) \) represents the time (in hours) the flight takes when the headwind speed is 100 miles per hour. Numerically, \( T(100)=\frac{300}{430}=\frac{30}{43}\approx0.698 \) hours.

Part b