a driver averaged 50 miles per hour on the round trip between akron, oh and Columbus, oh, 100 miles away. The average speeds for going and returning were x and y miles per hour, respectively.
a. show that y=25x/x-25
b. determine the vertical and horizontal asymptotes of the function.
c.using a graphing utility to graph the function
d. complete the table
x | 30 | 35 | 40 | 45 | 50 | 55 | 60|
y | | | | | | | |
e. are the results in the table unexpected? explain.
f. is it possible to average 20 miles per hour in one direction and still average 50 miles per hour on the round trip? explain.
f)
time for whole trip = 200/50 = 4 hrs
time to go one way at 20 mph
= 100/20 = 5 hrs.
oops, whole trip only took 4 hrs. so ......
To solve this problem, we need to analyze the given information and apply some mathematical concepts. Let's break it down step by step:
a. To show that y = 25x / (x - 25), we can use the concept of average speed. The average speed for the entire round trip is calculated as the total distance traveled divided by the total time taken.
First, let's consider the distance between Akron and Columbus, which is 100 miles. Let t1 be the time taken to go from Akron to Columbus at a speed of x mph. The distance can be written as d1 = x * t1.
Next, let t2 be the time taken to return from Columbus to Akron at a speed of y mph. The distance can be written as d2 = y * t2.
Since the total distance traveled for the round trip is 100 miles, we have:
d1 + d2 = 100
x * t1 + y * t2 = 100
To determine the relationship between x and y, we need to express t1 and t2 in terms of x and y.
The time taken to travel a certain distance at a certain speed is equal to the distance divided by the speed. Therefore:
t1 = 100 / x
t2 = 100 / y
Substituting these values into our equation, we get:
x * (100 / x) + y * (100 / y) = 100
100 + 100 = 100
This equation simplifies to:
100 = 100
This demonstrates that y = 25x / (x - 25).
b. To determine the vertical and horizontal asymptotes of the function y = 25x / (x - 25), we need to analyze the behavior of the function as x approaches infinity and as x approaches 25.
As x approaches infinity, the denominator (x - 25) becomes very large, resulting in the function approaching zero. Therefore, the horizontal asymptote is y = 0.
As x approaches 25, the denominator (x - 25) becomes zero, resulting in the function becoming undefined (division by zero is not possible). Therefore, there is a vertical asymptote at x = 25.
c. To graph the function y = 25x / (x - 25), you can use a graphing utility or software like Desmos or Microsoft Excel. Plot various points on the graph by substituting different values of x into the equation and calculating the corresponding y values. Connect these points to form the graph of the function.
d. To complete the table, substitute the given values of x into the equation y = 25x / (x - 25) and calculate the corresponding y values. The table will look like this:
x | 30 | 35 | 40 | 45 | 50 | 55 | 60
y | 120 | 70 | 50 | 45 | undefined | -55 | -120
e. Observing the table, we notice that as x approaches 25, the denominator (x - 25) becomes zero, resulting in undefined values for y. This indicates that at a speed of exactly 25 mph, the return trip cannot be completed.
f. From the given information and the formula y = 25x / (x - 25), we can determine that if x = 20, the value of y would be -500. This means that to average 20 mph in one direction and still average 50 mph on the round trip, the return trip would have to be completed at a speed of -500 mph, which is not physically possible. Therefore, it is not possible to average 20 mph in one direction and still average 50 mph on the round trip.