Question

Traveling 221 miles from Boston Back Bay Station to NYC Penn Station takes 3 hours

Answer 1

Answer:

Approximately $116\; \text{miles}$ for the train from Boston to NYC Penn Station.

Approximately $105\; \text{miles}$ for the train from NYC Penn Station to Boston.

Explanation:

Convert minutes to hours:

$\begin{aligned}t(\text{BOS \to NYC}) &= 3\; {\text{hour}} + 40\; \text{minute} \times \frac{1\; {\text{hour}}}{60\; \text{minute}} \\ &=\left(3 + \frac{2}{3}\right)\; \text{hour}\\ &= \frac{11}{3}\; \text{hour} \end{aligned}$.

$\begin{aligned}t(\text{NYC \to BOS}) &= 4\; {\text{hour}} + 5\; \text{minute} \times \frac{1\; {\text{hour}}}{60\; \text{minute}} \\ &= \frac{49}{15}\; \text{hour} \end{aligned}$.

Calculate average speed of each train:

$\begin{aligned}v(\text{BOS \to NYC}) &= \frac{s}{t}\\ &= \frac{221\; \text{mile}}{\displaystyle \frac{11}{3}\; \text{hour}} \\ &= \frac{663}{11}\; \text{mile} \cdot \text{hour}^{-1}\end{aligned}$.

$\begin{aligned}v(\text{NYC \to BOS}) &= \frac{s}{t}\\ &= \frac{221\; \text{mile}}{\displaystyle \frac{49}{15}\; \text{hour}} \\ &= \frac{2652}{49}\; \text{mile} \cdot \text{hour}^{-1}\end{aligned}$

Assume that it takes a time period of $t$ for the trains to pass by each other after departure. Distance each train travelled would be:

$s(\text{NYC \to BOS}) = v(\text{NYC \to BOS})\, t$.

$s(\text{BOS \to NYC}) = v(\text{BOS \to NYC})\, t$.

Since the trains have just passed by each other, the sum of the two distances should be equal to the distance between the stations:

$v(\text{NYC \to BOS})\, t + v(\text{BOS \to NYC})\, t = 221\; \text{mile}$.

Rearrange and solve for $t$:

$(v(\text{NYC \to BOS}) + v(\text{BOS \to NYC}))\, t = 221\; \text{mile}$.

$\begin{aligned}t &= \frac{221\; \text{mile}}{v(\text{NYC \to BOS}) + v(\text{BOS \to NYC})} \\ &= \frac{221\; \text{mile}}{\displaystyle \frac{663}{11}\; \text{mile} \cdot \text{hour}^{-1} + \frac{2652}{49}\; \text{mile} \cdot \text{hour}^{-1}} \\ &= \frac{539}{279}\; \text{hour}\end{aligned}$.

Distance each train travelled in $t = (539 / 279)\; \text{hour}$:

$\begin{aligned}s(\text{BOS \to NYC}) &= v\, t \\ &= \frac{663}{11}\; \text{mile} \cdot \text{hour}^{-1} \times \frac{539}{279}\; \text{hour} \\ &\approx 116\; \text{mile}\end{aligned}$.

$\begin{aligned}s(\text{NYC \to BOS}) &= v\, t \\ &= \frac{2652}{49}\; \text{mile} \cdot \text{hour}^{-1} \times \frac{539}{279}\; \text{hour} \\ &\approx 105\; \text{mile} \end{aligned}$.