Question

Average Speed On a trip of \(d\) miles to another city, a truck driver's average speed was \(x\) miles per hour. On the return trip. the average speed was \(y\) miles per hour. The average speed for the round trip was 50 miles per hour. (a) Verify that \(y=\frac{25 x}{x-25}\) What is the domain? (b) Complete the table. \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 30 & 40 & 50 & 60 \\ \hline\(y\) & & & & \\ \hline \end{tabular} Are the values of \(y\) different than you expected? Explain. (c) Find the limit of \(y\) as \(x \rightarrow 25^{+}\) and interpret its meaning.

Short answer

The domain of \( y \) is \( x \in \mathbb{R}, x \neq 25 \). After filling the table, the corresponding values of \( y \) are \( 150, 66.67, 50, 42.86 \), respectively. The limit of \( y \) as \( x \rightarrow 25^{+} \) is infinity.

Step-by-step solution

Verify the Function and Find the Domain

For part (a), one needs to substitute the given conditions into the formula of average speed to verify the function \( y = \frac{25x}{x-25} \). To find the domain, one should realize that all real numbers are permissible for \( x \) except the one that makes the denominator zero, that is \( x \neq 25 \). Therefore, the domain is \( x \in \mathbb{R}, x \neq 25 \)

Complete the Table

For part (b), substitute the given values of \( x \) into the formula to compute the values of \( y \). The corresponding values of \( y \) would be \( \frac{25(30)}{30-25}=150 \), \( \frac{25(40)}{40-25}=66.67 \), \( \frac{25(50)}{50-25}=50 \), and \( \frac{25(60)}{60-25}=42.86 \), respectively. The results of \( y \) may be different than expected if one doesn't consider the counterintuitive nature of harmonic averages.

Find the Limit and Interpret Its Meaning

For part (c), finding the limit as \( x \rightarrow 25^{+} \) can be done directly by substituting into the formula. The limit yields infinity (\( \lim_{x\to 25^+} \frac{25x}{x-25} = \infty \)). This means that as the average speed for the trip to the city approaches 25 mph from the right, the average speed for the return trip tends to infinity. This is because as the speed decreases, the time taken for the trip increases.