Question

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. $$ \begin{array}{|c|c|c|c|c|}\hline x & {30} & {40} & {50} & {60} \\ \hline y & {} & {} & {} \\ \hline\end{array} $$ Are the values of \(y\) different than you expected? Explain. (c) Find the limit of \(y\) as \(x\) approaches 25 from the right and interpret its meaning.

Short answer

The domain of \(y\) is \(x ≠ 25\), because at \(x = 25\), the equation gives an undefined value. The table results for \(y\) are 37.5, 66.67, 125 and 300. The limit of \(y\) as \(x\) approaches 25 from the right is infinity, indicating an extremely high (or even infinite) speed requirement for maintaining the average speed of 50 mph.

Step-by-step solution

Verification

To verify the equation \(y=\frac{25 x}{x-25}\), we know that the total distance is \(2d\) (to the other city and back), and the total time is \(d/x + d/y\). Since average speed is total distance over total time, we have \(50 = \frac{2d}{d/x + d/y}\). After simplifying, we find that \(y=\frac{25 x}{x-25}\). Therefore, the equation is verified.

Finding the Domain

The domain of \(y\) is the set of all possible values of \(x\). In this case, it is all real numbers except for 25, because if \(x=25\), then the denominator of the equation becomes 0 and the expression is undefined.

Completing the table

Substitute \(x = 30, 40, 50, 60\) into the equation \(y=\frac{25 x}{x-25}\). The corresponding y values are 37.5, 66.67, 125, and 300. This indicates that as \(x\) increases, \(y\) increases much faster which may be counter-intuitive initially as we expect both \(x\) and \(y\) to increase linearly.

Finding the limit value

In calculus, we need to find the limit of \(y\) as \(x\) approaches 25 from the right, denoted as \(\lim_{x \to 25+} y\). According to the equation, the closer \(x\) is to 25, the closer \(y\) will approach infinity. That means if the driver's average speed on either leg of the trip is too close to the round trip average speed, then the average speed on the other leg must be extremely high, or even infinite to maintain the overall method.

Key concepts

Limits and Continuity

When exploring the behavior of functions in calculus, two concepts that go hand in hand are limits and continuity. These two concepts help us to understand how a function behaves near certain points, especially when it can't be evaluated directly at those points.

In the case of the truck driver's problem, we are particularly interested in the limit of the function as the input approaches a value from one side, known as a one-sided limit. Here, we examine the average speed function given by the equation \(y = \frac{25x}{x-25}\), as the average speed \(x\) on the outward trip approaches 25 from the right. As \(x\) gets closer to 25, the denominator of our function approaches zero, causing the value of \(y\) to skyrocket towards infinity.

A function is said to be continuous at a point if the limit as you approach that point exists and is equal to the function's value at that point. In this exercise, since \(y\) becomes infinite as \(x\) approaches 25, the function is not continuous at \(x = 25\). This particular discontinuity is known as a vertical asymptote, which is a line that a function approaches but never touches or crosses.

Rate of Change

The rate of change in a mathematical context often refers to how one quantity changes in relation to another. In our average speed problem, rate of change provides insight into how the return trip's speed \(y\) changes with respect to the departure speed \(x\).

In calculus, the rate of change is represented by the derivative of a function. However, even without using derivatives, we can detect patterns of change by observing the values computed for the function. From the completed table in the problem's solution, we notice that as \(x\) increases, \(y\) increases disproportionately. This demonstrates a non-linear relationship where the rate of change between \(x\) and \(y\) is not constant; it increases as \(x\) moves further away from 25. The function \(y = \frac{25x}{x-25}\) has a rate of change that illustrates the sensitivity of the truck driver's return trip speed to changes in the outward trip speed—particularly significant when the outward speed is near the round trip average speed of 50 miles per hour.

Domain of a Function

The domain of a function is the set of all possible input values (typically represented by \(x\)) for which the function is defined and returns a real number. In many cases, the domain will consist of all real numbers, but there are situations where the function is not defined for certain inputs, which are then excluded from the domain.

In the textbook problem we're examining, the formula \(y = \frac{25x}{x-25}\) is undefined when \(x = 25\) because it would require division by zero, which is not permitted in mathematics. Therefore, the domain of this function includes all real numbers except \(x = 25\). Understanding the domain is crucial when evaluating functions and understanding their behavior across various input values.

From a practical standpoint, recognizing restrictions on the domain helps prevent misinterpretation of a situation. Had we not understood the domain of our truck driver's speed function, we might have incorrectly assumed that an average speed on the outward trip could be 25 miles per hour, which would lead to the irrational conclusion of an infinite return speed to maintain an average of 50 miles per hour for the round trip.