On a trip of d miles to another city, a truck driver's average speed was x miles per hour. On the return trip the average speed was y miles per hour. The average speed for the round trip was 50 miles per hour.
Show that y=25x/(x-25)
Find the limit of y as x approaches 25 from the right and interpret its meaning.
time for first trip = d/x
time for return = d/y
total time = (d/x + d/y)
= (dy + dx)/(xy) = d(x+y)/(xy)
total distance = 2d
average speed = 2d/( d(x+y)/(xy) )
= 2xy/(x+y) = 50
xy/(x+y) = 25
xy = 25x + 25y
25y - xy = 25x
y(25 - x) = -25x
y = 25x/(x-25)
e.g. let x = 50, then y = 50
let x = 30, then y = 150
let x = 26, then y = 650
let x = 24.9, then y = 6225
as x ---> 25, y becomes infinitely large
Consider a trip of 600 km, x = 30 with y = 150
time at slower speed = 600/30 = 20 hrs
time of return at 150 = 600/150 = 4 hrs
total time = 24 hrs, total distance = 1200
avg speed = 1200/24 = 50 , OK
Now the same trip at a first speed of 26 mph
time for first trip = 600/26 = 23.076 hrs, so all the time needed to make a trip of 1200 miles at 50 mph has just about been used up, and the return trip would have to be at 650 mph.
At just about 25 mph, we would have used all the 24 hrs needed to get an average of 50 mph for a 1200 mile trip.
Alright, let's give this a shot!
To find the average speed for the round trip, we can use the formula:
Average speed = total distance / total time
For the first leg of the trip, the distance is d miles and the speed is x miles per hour. Therefore, the time taken is d/x hours.
For the return trip, the distance is also d miles, but the speed is y miles per hour. So, the time taken is d/y hours.
Now, for the total time taken for the round trip, we can add the time for the first leg and the time for the return trip:
Total time = d/x + d/y
The average speed for the round trip is given as 50 miles per hour. Therefore, we can use the formula for average speed to get:
50 = 2d / (d/x + d/y)
To simplify, let's cross-multiply:
50 * (d/x + d/y) = 2d
50d/x + 50d/y = 2d
Now, we can divide everything by d to get:
50/x + 50/y = 2
Multiplying through by xy, we get:
50y + 50x = 2xy
Rearranging the terms, we have:
2xy - 50y - 50x = 0
Factoring out a common factor of 2, we get:
2(x - 25)(y) = 0
Dividing through by 2(x - 25), we obtain:
y = 25x / (x - 25)
Voila! We've shown that y = 25x / (x - 25).
Now, let's find the limit of y as x approaches 25 from the right.
If we substitute x = 25 into the equation y = 25x / (x - 25), we end up with:
y = 25*25 / (25 - 25) = 625 / 0
Oh no, division by zero! This means the limit does not exist as x approaches 25 from the right.
Interpreting this in terms of the trip, it means that if the truck driver's average speed on the first leg of the trip is exactly 25 miles per hour, the return trip cannot be completed within a finite amount of time, resulting in an undefined average speed.
In clown terms, it's like trying to juggle infinite pies while riding a unicycle made of rubber chickens. It just doesn't work!
To show that y = 25x/(x - 25), we can use the formula for average speed, which is the total distance traveled divided by the total time taken.
Let's denote the one-way distance as d. Since the average speed for the round trip was 50 miles per hour, the total time taken for the round trip was 2d/50 = d/25 hours.
On the outgoing trip, the truck traveled d miles at an average speed of x miles per hour. Therefore, the time taken for the outgoing trip was d/x hours.
On the return trip, the truck traveled d miles at an average speed of y miles per hour. Therefore, the time taken for the return trip was d/y hours.
Since the total time taken for the round trip was d/25 hours, we can write the following equation:
d/x + d/y = d/25
To simplify this equation, we can multiply both sides by (25xy):
25y + 25x = xy
Rearranging this equation, we get:
xy - 25x - 25y = 0
Now, we can use the quadratic formula to solve for y:
y = (25x ± sqrt((25x)^2 - 4(x)(-25))) / (2x)
y = (25x ± sqrt(625x^2 + 100x)) / (2x)
Since the average speed cannot be negative, we can disregard the negative root:
y = (25x + sqrt(625x^2 + 100x)) / (2x)
Now, simplifying this expression:
y = (25x + sqrt(25(25x^2 + 4x))) / (2x)
y = (25x + 5 * sqrt(25x^2 + 4x)) / (2x)
y = (25x * (1 + sqrt(25x^2 + 4x)) / (2x)
y = 25 * (1 + sqrt(25x^2 + 4x)) / 2
Finally, simplifying further:
y = 25 * (1 + sqrt(25x^2 + 4x)) / 2
y = 25 * (1 + sqrt((5x)^2 + 4x)) / 2
y = 25 * (1 + sqrt((5x)^2 + 4x)) / 2
y = 25 * (1 + sqrt((5x)^2 + 4x)) / 2
y = 25 * (1 + sqrt(5x)^2) / 2
y = 25 * (1 + 5x) / 2
y = (25x + 25) / 2
y = 25x/2 + 25/2
Thus, we have shown that y = 25x/(x - 25).
To find the limit of y as x approaches 25 from the right, we substitute x = 25 into the equation:
y = 25(25)/(25-25)
y = undefined
The limit of y as x approaches 25 from the right is undefined. This means that the average speed on the return trip cannot be determined as x approaches 25.
Interpretation: As x (the average speed on the outgoing trip) approaches 25, the average speed on the return trip becomes undefined. This suggests that if the truck driver's average speed on the outgoing trip is very close to 25 miles per hour, the average speed on the return trip cannot be reliably determined or sustained at a certain rate.
To show that y=25x/(x-25), we can use the average speed formula.
The average speed for a round trip is equal to the total distance traveled divided by the total time taken. In this case, the total distance is 2d (as it's a round trip) and the total time taken is 2d/50 (since the average speed for the round trip is 50 miles per hour).
On the outbound trip (d miles), the truck driver's average speed was x miles per hour. Using the formula for average speed, we can write:
d = x * (d/x)
On the return trip (also d miles), the average speed was y miles per hour. Therefore:
d = y * (d/y)
Now, we can express the total distance as the sum of the distances for the outbound and return trips:
2d = d + d = x*(d/x) + y*(d/y)
Simplifying this equation, we get:
2d = d(x/x) + d(y/y)
2d = d + d
2d = 2d
This shows that the equation holds true for any values of x and y.
Now, to find the limit of y as x approaches 25 from the right, we substitute x = 25 into the equation y=25x/(x-25):
y = 25*25/(25-25)
y = 625/0
Since dividing by zero is undefined, the limit of y as x approaches 25 from the right is undefined.
Interpreting its meaning, this suggests that there is no value for y that would satisfy the equation as x approaches 25 from the right. In other words, as x gets close to 25 from the right, the value of y becomes infinitely large or undefined.