\( 1000 \mathrm{~km} \)
\[
\frac{1000 \mathrm{~km}}{\operatorname{man} \text { windspeed } \frac{\omega}{\omega} \mathrm{km} / \mathrm{m}}
\]
1. An airplane makes a round trip where the one-way distance is \( 1000 \mathrm{~km} \). On the outleg the plane faces a headwind of \( w \mathrm{~km} / \mathrm{hr} \), while on the return there is a tailwind of \( w \mathrm{~km} / \mathrm{hr} \). The speed of the plane in still air is \( 400 \mathrm{~km} / \mathrm{hr} \).
a. Show that the function
\[
T(w)=\frac{800,000}{160,000-w^{2}}
\]
expresses the total time for the trip (in hours) as a function of the wind speed \( w \) (in \( \mathrm{km} / \mathrm{hr} \) ). (3 pts.)
rate time \( = \) distance, so
\[
(400-w) t_{1}=1000 \text { and }(400+w) t_{2}=1000
\]
where \( t_{1} \) is time for outboud log \( \phi \)
\( t_{2} \) is time for return.
\[
t_{1}=\frac{1000}{400-w} \text { and } t_{2}=\frac{1000}{400+w}
\]
total time \( T=t_{1}+t_{2}=\frac{1000}{400-\omega}+\frac{1000}{1001 \omega} \)
\[
\begin{array}{l}
T= \frac{1000(400+\omega)+1000(400-\omega)}{(100-\omega)(400+\omega)} \\
T=\frac{800,000}{160000-\omega^{2}} \\
\text { b. Give the domain for } T(\omega) \text { that makes sense in the context of this prob }
\end{array}
\]
b. Give the domain lof \( T(w) \) that makes sense in the context of this problem. Explain your reasoning. ( 2 pts.)
The proper domain is \( \omega \in(-400,400) \)
If \( w=0 \), there is no wind, whick is fine. If \( w \in(0,400) \), then the problen will make sense becauge the wind speed is (ess than the plare speed.
If \( \omega \in(-400,0) \), the the wing is joing the opporite way cempard to the language in the pioslom buit still thectinicaly okas. If \( w \geq 400 \) or \( w \leq-400 \), the wind faster the plane' set to s. the plane cannot' set destincs.
c. Sketch a graph of \( T(w) \) including intercepts and asymptotes. (2 pts)
d. Describe the meaning of important elements of the graph in part (c) relative to the context of the problem. (3 pts.)
(1) At \( (0,5) \) ther is no wind, so the plane takes 5 howr
