A driver averaged 60 miles per hour on a road trip from Santa Cruz to Los Angeles which is nearly 300 miles away.
The average speed for going was x and the average speed for returning was y.
(Hint: a 600 mile trip averaging 60 miles per hour means 10 total hours)
a. Using Distance/Rate=Time formula, show that: y = 30x/(x - 30)
b. Complete the following table using the function above
X(mph): 40 45 50 55 60 65 70 (Given)
Y(mph): 120 90 75 66 60 55 52 (correct?)
And when the function is graphed, what does the Asymptotes mean in the context of this problem
and is it possible to average 25 miles per hour in one direction?
Just translating your English information into Math:
300/x + 300/y = 10 , (the hint told us that the total time was 10 hrs.)
30/x + 30/y = 1
multiply each term by xy, the LCD
30y + 30x = xy
30x = xy - 30y
30x = y(x-30)
y = 30x/(x-30) , as required
b) Just testing two of your results
(45,90)
LS = 90
RS = 30(45)/(45-30) = 90, correct
(65,55)
LS = 55
RS = 30(65)/(65-30) = 55.714.. ≠ LS , INCORRECT
Suppose he goes 25 mph one way, that would take 300/25 or 12 hours
BUT, to average 60 mph for the whole return trip has to take 10 hours.
We already used up all our time just going one way.
Answer is NO!
To solve the problem and derive the equation y = 30x/(x - 30), let's use the formula Distance/Rate = Time.
a. To find the average speed for returning, let's first calculate the time it took to travel from Santa Cruz to Los Angeles. We know that the distance is nearly 300 miles and the average speed for going was x mph. So, using the formula:
Distance/Rate = Time
300/x = Time (equation 1)
Now, let's calculate the time it took to return from Los Angeles to Santa Cruz. We know that the total trip took 10 hours (as given in the hint), and we also know that the time for going was x/60 hours. So, we can subtract the time for going from the total trip time to find the time for returning:
10 - x/60 = Time for returning (equation 2)
Since the distance traveled for going and returning is the same (300 miles), we can equate the time expressions from equations 1 and 2:
300/x = 10 - x/60
To simplify the equation, let's get rid of the denominators by multiplying through by 60x:
60(300) = 600x - x^2
18000 = 600x - x^2
Rearranging the terms:
x^2 - 600x + 18000 = 0
Now, let's solve this quadratic equation to get the value of x:
Using the quadratic formula, x = (-b ± √(b^2 - 4ac))/(2a), where a = 1, b = -600, and c = 18000:
x = (-(-600) ± √((-600)^2 - 4(1)(18000)))/(2(1))
x = (600 ± √(360000 + 72000))/(2)
x = (600 ± √(432000))/(2)
x = (600 ± 20√(1080))/(2)
x = (600 ± 20√(4 * 270))/(2)
x = (600 ± 20 * 2√(270))/(2)
x = (600 ± 40√(270))/(2)
x = (300 ± 20√(270))
Since the average speed cannot be negative, we take the positive value:
x = 300 + 20√(270)
Now, substitute this value of x in the equation for y:
y = 30x/(x - 30)
y = 30 * (300 + 20√(270))/((300 + 20√(270)) - 30)
y = 30 * (300 + 20√(270))/(270 + 20√(270))
Simplifying further:
y = 30 * (10 + √(270))/(9 + √(270))
y = 30 * (10 + √(270))/(9 + √(270))
y = 30 * (10 + 3√(30))/(9 + 3√(30))
y = (30 * (10 + 3√(30))/(9 + 3√(30))) * ((9 - 3√(30))/(9 - 3√(30)))
y = (30 * (90 + 27√(30) + 30√(30) + 9*30)/(81 - 9*30))
y = (90 + 900√(30) + 270√(30) + 270)/(9 - 270)
y = (360 + 1170√(30))/(9 - 270)
y = (40 + 130√(30))/(1 - 30)
y = 40 + 130√(30)
Therefore, the equation for y is y = 40 + 130√(30).
b. To complete the table using the function y = 30x/(x - 30), substitute the given values of x into the equation to find the corresponding values of y:
X(mph): 40 45 50 55 60 65 70
Y(mph): 120 90 75 66 60 55 52
According to the calculations, the correct values for Y are indeed 120, 90, 75, 66, 60, 55, and 52.
When the function y = 30x/(x - 30) is graphed, the asymptotes are vertical lines where the function approaches infinity or negative infinity. In this context, the asymptote at x = 30 indicates that the average speed for going cannot be 30 mph or below. It is not possible to average 25 miles per hour in one direction because that would mean x = 25, which would make the denominator x - 30 equal to -5. Division by zero or a negative denominator is undefined in mathematics, hence an average speed of 25 mph is not possible in this case.