Word problem: express time traveled as a rate of the bike, graph and confirm algebraicaly

Question

Josh rode his bike 17 miles from his home in Columbus and then traveled 53 miles by car from Columbus to Dayton.  Assume average rate of the car was 43 miles faster than average rate of bike. 

 

Express the the total time required to complete the 70 miles as a function of the rate x of the bike. 

 

Find b's graphically the rate of the bike if the total time is 1hr 40 min. Confirm algebraicaly. 

Answer 1

To understand how to solve this question, you first have to figure out what "x" really represents. The problem states that "x" is the rate of the bike, which is most easily expressed in miles per hour. You need to express time as a function of "x", so "y" must be time. Let's make it time in hours.

 

So, to define the variables:

let x = rate of the bike in miles per hour

let y = time in hours

 

How does time relate to rate? Well, if you go 100 miles at 50 miles per hour, it takes 2 hours. Traveling 75 miles at 30 mph takes 2.5 hours. You need to divide the distance by the rate to get the time. You can see this algebraically by the formula distance=rate*time, or d=rt. To solve for "t", divide both sides by "r".

 

Notice that there are two rates in the problem, and we are given the distance for each rate. The total travel time (y) is equal to the sum of the bike and car travel times. The bike travel time is the bike distance over the bike rate, or 17/x. The car travel time is the car distance over the car rate, or 53/(x+43), since the car rate is 43 mph faster than the bike rate (x).

 

So y=17/x + 53/(x+43).

 

This is the graph of a rational function. It should look like to hyperbolas with a vertical and a horizontal asymptote. If your graph is off, check that you have parentheses in the correct places.

 

Now you need to find the rate if the time is 1hr 40min, or 1 and 2/3 hours, or 5/3hrs. Remember that time is y. If you are using a graphing calculator, and y₁ is your first graph, put y₂=5/3 for the second, then calculate the intersection (on the positive side, since rate is positive). You may need to increase your viewing window.