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.
a. To show that y = 25x / (x - 25), we can use the formula for average speed, which is total distance divided by total time.
Let the distance between Akron and Columbus be 100 miles. The time taken to travel from Akron to Columbus is given by distance / speed, which is 100 / x. The time taken to return from Columbus to Akron is given by distance / speed, which is 100 / y.
The total time for the round trip is the sum of the time taken to go and the time taken to return: (100 / x) + (100 / y).
The average speed for the round trip is the total distance divided by the total time:
Average speed = (2 * 100) / [(100 / x) + (100 / y)].
We are given that the average speed for the round trip is 50 mph, so we can substitute this into the equation above:
50 = (2 * 100) / [(100 / x) + (100 / y)].
Simplifying this equation, we get:
50 * [(100 / x) + (100 / y)] = 200.
[(100 / x) + (100 / y)] = 4.
Now, we can solve this equation for y:
(100 / x) + (100 / y) = 4.
100 / y = 4 - (100 / x).
100 / y = (4x - 100) / x.
y / 100 = x / (4x - 100).
y = (100x) / (4x - 100).
y = 25x / (x - 25).
Therefore, y = 25x / (x - 25).
b. To determine the vertical and horizontal asymptotes of the function, we need to analyze the behavior of y as x approaches positive or negative infinity.
The vertical asymptote occurs when the denominator of y becomes zero, resulting in an undefined value. In this case, the denominator is (x - 25), so x = 25 is the vertical asymptote.
The horizontal asymptote occurs when the degree of the numerator and the denominator is the same. In this function, the numerator (25x) and the denominator (x - 25) both have a degree of 1. Therefore, the horizontal asymptote is y = (25 * 1) / (1 - 0) = 25.
c. Using a graphing utility, graph the function y = 25x / (x - 25). This can be done by inputting the equation into a graphing calculator or online graphing tool. The resulting graph will show the shape and behavior of the function.
d. To complete the table, we can substitute different values of x into the equation y = 25x / (x - 25) and calculate the corresponding values of y.
x | 30 | 35 | 40 | 45 | 50 | 55 | 60
y | 125 | 87.5 | 62.5 | 47.5 | 37.5 | 31.8 | 27.3
By substituting x values into the equation, we determined the corresponding values for y.
e. The results in the table show that as x approaches 25 (the vertical asymptote), y approaches positive infinity. This can be seen as x gets closer and closer to 25, the values of y become larger and larger.
f. It is not possible to average 20 miles per hour in one direction and still average 50 miles per hour on the round trip. This is because the equation y = 25x / (x - 25) shows that as x approaches 25 (the vertical asymptote), the value of y approaches infinity. Therefore, if x = 20, the value of y would also approach infinity, meaning that the average speed for the round trip would not be 50 miles per hour.