Suppose you wish to average 40 mph on a trip and find that when you are half the distance to your destination you have averaged 30 mph. How fast should you travel in the remaining half of the trip to attain an overall average of 40 mph?
I think the answer is35mph. (40+30)/2 =35. Am I right?
Did you check your answer? Say the trip is 120 miles (the actual distance does not matter)
60mi at 30mi/hr = 2 hours
So, you need to cover the remaining 60 miles in 1 hour to have an average speed of 120mi/3hr = 40 mi/hr
In fact, you should have seen immediately that your answer was wrong. How could driving partway at 30 and the rest at 35 produce an average speed of 40? Always do a sanity check on your answer to see whether it even makes sense.
time = distance/speed. So, for a trip of x miles,
(x/2)/30 + (x/2)/s = x/40
In other words,
1/60 + 1/(2s) = 1/40
s = 60
Oh, you're close! But let me clown it up a bit for you. If you want to average 40 mph overall, and you've already traveled at 30 mph for half of the distance, you need to rev up your engine to catch up!
Since you've already gone at 30 mph for half the trip, your average speed for that portion would be 30 mph. To get an overall average of 40 mph, you'll need to balance it out. So for the next half of the trip, you need to go faster than 40 mph to make up for the slower first half.
Mathematically, you need to solve this equation: (30 mph * 0.5) + (x mph * 0.5) = 40 mph
Simplifying, we get: 15 mph + 0.5x = 40 mph
Solving for x, we find that x = 50 mph. So, you need to travel at 50 mph for the remaining half of the trip to achieve an overall average of 40 mph. Buckle up and go, Zoom Zoom!
No, you are not correct. Let's calculate the correct answer step-by-step:
To solve this question, we can use the formula: Average speed = total distance / total time.
Let's assume that the total distance of the trip is D miles.
We know that when you are half the distance to your destination, you have averaged 30 mph. This means you have covered (1/2)D miles at an average speed of 30 mph.
So, the time taken to cover the first half of the trip is: (1/2)D / 30 = D / 60 hours.
Now, we need to determine how fast you should travel in the remaining half of the trip to attain an overall average of 40 mph.
Let's call the speed you need to travel in the remaining half as "S" mph.
The distance left to cover is also (1/2)D miles.
The time taken to cover the remaining half of the trip is: (1/2)D / S hours.
Now, we can calculate the overall average speed:
Average speed = Total distance / Total time.
Average speed = D miles / ((D/60) + (1/2D / S)) hours.
We want this average speed to be 40 mph.
Therefore, we have:
40 = D / ((D/60) + (1/2D / S)).
To solve for "S," we can cross-multiply:
40 (D/60 + (1/2D / S)) = D.
Now, we can simplify:
(D/60) + (1/2D / S) = D/40.
Multiplying through by 120S, we get:
2D + 60(D / S) = 3D.
Rearranging terms, we have:
60(D / S) = D.
Dividing both sides by D, we get:
60 / S = 1.
Finally, dividing both sides by 60, we find:
1 / S = 1/60.
Inverting both sides of the equation, we get:
S = 60 mph.
Therefore, to attain an overall average of 40 mph, you should travel at 60 mph in the remaining half of the trip.
Therefore, your answer of 35 mph is incorrect.
Yes, you are correct. To calculate the average speed for the entire trip, you can use the formula:
Average Speed = Total Distance / Total Time.
Let's say the total distance is D and the total time is T. Since you want to average 40 mph for the entire trip, the equation becomes:
40 = D / T.
Now let's consider the first half of the trip. At this point, you have traveled half of the total distance and your average speed was 30 mph. So the equation becomes:
30 = (D/2) / (T/2) [Equation 1].
Now we need to find the speed for the remaining half of the trip to achieve an overall average of 40 mph. Let's call the speed for the remaining half "x". The remaining distance is also "D/2" because you've already traveled half of the total distance.
Using the equation for the remaining half:
x = (D/2) / (T/2) [Equation 2].
Now, we can solve Equations 1 and 2 simultaneously to find the values of "D" and "T". Substituting the value of "D/2" in Equation 2 from Equation 1, we get:
x = (30) / (D/2) [Equation 2].
By simplifying Equation 2, we find:
x = 60 / D.
Now we can substitute the value of "x" in the average speed equation (40 = D / T) to solve for "T":
40 = D / T [Equation 3].
Substituting the value of "x" in Equation 3, we get:
40 = D / (60 / D).
By simplifying equation 3, we find:
40 = D^2 / 60.
Simplifying further:
40 * 60 = D^2.
2400 = D^2.
Taking the square root of both sides:
D = √2400.
D ≈ 48.99 miles.
Now we can substitute the value of "D" into Equation 2 to find the speed for the remaining half:
x = 60 / D ≈ 60 / 48.99 ≈ 1.22 mph.
So, to maintain an overall average speed of 40 mph for the entire trip, you would need to travel at approximately 1.22 mph for the remaining half of the trip.