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Chapter 1: Problem 46

A commuter rides his bicycle to the train station, takes the subway downtown and then walks from the subway station to his office. He bikes at an average speed of \(B\) miles per hour and can walk \(M\) miles in \(H\) hours. The subway ride takes \(R\) minutes. The commuter bikes \(X\) miles and walks \(Y\) miles to get to work. Assume that \(X, Y, B, H\), \(M\), and \(R\) are all constants. The amount of time it takes the commuter to get to work varies with how long he has to spend at the subway station locking his bike and waiting for the next train. Denote this time by \(w\), where \(w\) is in hours. Express the time it takes for his commute as a function of \(w\). Specify whether your answer is in minutes or in hours.

Short Answer

Expert verified
The total time for the commuter's journey, in hours, is given by \(X/B + R/60 + YH/M + w\).

Step by step solution

01

Establish Time for Bicycling

To begin, let's consider the time taken by the commuter to bike to the train station. We know that Time = Distance/Speed, so in this case, the time (in hours) taken to bike to the train station is \(X/B\).
02

Establish Time for Subway Ride

We know that the subway ride takes \(R\) minutes. To standardize our units, we'll need to convert this to hours. As there are 60 minutes in an hour, the subway ride takes \(R/60\) hours.
03

Establish Time for Walking

The speed at which the commuter walks is described as \(M\) miles in \(H\) hours, or \(M/H\) miles per hour. Therefore, the time the commuter spends walking from the subway station is \(Y/(M/H)\), which simplifies to \(YH/M\).
04

Calculate Total Time Spent

To calculate the total time spent on the commute, we simply sum up all the individual times - bicycling, subway ride, walking, and waiting at the subway station. The total time spend on commute is therefore \((X/B) + (R/60) + (YH/M) + w\), where all time measurements are in hours.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Speed
The concept of average speed is fundamental when dealing with commute times. Average speed is simply the total distance traveled divided by the total time taken to travel that distance. In mathematical terms, the average speed (v_{avg}) is given by the formula:
\[ v_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} \]
When we apply this to our commuter's bicycling and walking speeds, we have the average speed for biking as \(B\) miles per hour and walking as the ratio of \(M\) miles to \(H\) hours, or \(M/H\) miles per hour. It's essential to stick to consistent units across the calculations, as mixing miles per hour with miles in hours can lead to confusion.

Additionally, it's helpful to recognize that average speed can vary during different segments of the journey and understanding each segment 鈥� biking, subway, and walking 鈥� allows us to create a complete picture of the commute.
Time-Distance Relationship
Understanding the time-distance relationship is crucial when calculating commute times. This relationship shows how time and distance are inversely related; when speed is constant, the time taken to cover a certain distance increases as the distance increases and vice versa.

An essential formula for this concept is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
For our commuter, the time to bike \(X\) miles is \(X/B\) hours and to walk \(Y\) miles is \(YH/M\) hours. And let's not forget the additional waiting time \(w\), which is part of the total commute time but does not correspond to covering any distance. Applying this principle helps students visualize how each part of the journey contributes to the overall time spent commuting.
Rate of Change
Finally, the rate of change is a concept that refers to how a quantity changes with respect to another. In the context of commuting, the rate of change can refer to how the commuter's travel time changes as his waiting time at the subway station varies. In calculus, the rate of change is the derivative of a function. While we aren't differentiating here, understanding the principle can help predict changes in commute time based on changes in one part of the journey.

In simpler terms, if the commuter's waiting time at the subway station \(w\) increases, then the total commute time increases at the same rate, because all other factors of the journey remain constant. Apply this concept to the commuter's scenario helps in appreciating how slight adjustments to any leg of the commute can either shorten or lengthen the overall time.

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Most popular questions from this chapter

Some friends are taking a long car trip. They are traveling east on Route 66 from Flagstaff, Arizona, through New Mexico and Texas and into Oklahoma. Let \(f\) be the function that gives the number of miles traveled \(t\) hours into the trip, where \(t=0\) denotes the beginning of the trip. For instance, \(f(7)\) is the mileage 7 hours into the trip. If the travelers set an odometer to zero at the start of the trip, the output of \(f\) would be the reading on the odometer. Let \(g\) be the function that gives the car's speed \(t\) hours into the trip, where \(t=0\) denotes the beginning of the trip. For instance, \(g(7)\) is the car's speed 7 hours into the trip. The output of \(g\) corresponds to the speedometer reading. Suppose they pass a sign that reads "entering Gallup, New Mexico," \(h\) hours into the trip. (a) Write the following expressions using functional notation wherever appropriate. i. The car's speed 1 hour before reaching Gallup ii. 10 miles per hour slower than the speed of the car entering Gallup iii. Half the time it took to reach Gallup iv. Their speed 6 hours after reaching Gallup v. The distance traveled in the first 2 hours of the trip vi. The distance traveled in the second 2 hours of the trip vii. Half the distance covered in the second 3 hours of travel viii. The average speed in the first 5 hours of travel (Average speed is computed by dividing the distance traveled by the time elapsed.) ix. The average speed between hour 6 of the trip and hour 12 of the trip (b) Interpret the following in words. i. \(f(h+2)\) ii. \(\frac{1}{2} f(h)\) iii. \(f\left(\frac{h}{2}\right)\) iv. \(f(h-2)\) v. \(f(h)-2\) vi. \(f(h)+2\) vii. \(g(h+2)\) viii. \(g(h)+2\) ix. \(g(h)-2\) x. \(\frac{1}{2} g(h)\) xi. \(\frac{1}{2} g(h-1)\)

Writing: We would like to tailor this course to your needs and interests; therefore we'd like to find out more about what these needs and interests are. On a sheet of paper separate from the rest of your homework, please write a paragraph or two telling us a bit about yourself by addressing the following questions. (a) What are you interested in studying in the future, both in terms of math and otherwise? (b) Are there things you have found difficult or confusing in mathematics in the past? If so, what? (c) What was your approach to studying mathematics in the past? Did it work well for you? (d) What are your major extracurricular activities or interests? (e) What do you hope to get out of this course?

(a) find the value of the function at \(x=0, x=1\), and \(x=-1\). (b) find all \(x\) such that the value of the function is \((i) 0,(i i) 1\), and \((i i i)-1 .\) $$ g(x)=x^{2}-1 $$

(a) A bead maker has a collection of wooden spheres 2 centimeters in diameter and is making beads by drilling holes through the center of each sphere. The length of the bead is a function of the diameter of the hole he drills. Find a formula for this function. If you are stuck, begin by trying to express half the length of the bead as a function of the radius of the hole drilled. (b) More generally, suppose he works with spherical beads of radius \(\mathrm{R}\). Again, the length of the bead is a function of the diameter of the hole he drills. Find a formula for this function. In the following problems, demonstrate your use of the portable strategies for problem solving described in this chapter. What simpler questions are you asking yourself? What concrete example can you give to convince your friends and relatives that you are right? Write this up clearly, so a reader can follow your train of thought easily.

Two sisters, Nina and Lori, part on a street corner. Lori saunters due north at a rate of 150 feet per minute and Nina jogs off due east at a rate of 320 feet per minute. Assuming they maintain their speeds and directions, express the distance between the sisters as a function of the number of minutes since they parted.
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