a man rides up a ski lift at the rate of 5 miles per hour and skis back down the hill at the rate of 50miles an hour. if the complete trip requires 22 minutes, how far is it from the bottom to top of the hill?
22 minutes = 22/60 or 11/30 hours.
let time going up be t hrs
then time going down is (11/30 - t) hrs.
5t = 50(11/30 - t)
5t = 55/3 - 50t
55t = 55/3
t = 1/3
distance to top is 5t = 5(1/3) = 5/3 miles
To find the distance from the bottom to the top of the hill, we can use the formula:
Distance = Speed × Time
Let's break down the given information:
The man rides up the ski lift at a rate of 5 miles per hour.
The man skis back down the hill at a rate of 50 miles per hour.
The complete trip requires 22 minutes.
First, let's convert the time to hours. Since there are 60 minutes in 1 hour, we have:
22 minutes ÷ 60 minutes/hour = 0.367 hours
Next, let's calculate the time it takes for the man to ride up the ski lift:
Time riding up = distance ÷ speed riding up
Let the distance from the bottom to the top of the hill be "d".
So, we have:
d/5 miles per hour
Now, let's calculate the time it takes for the man to ski back down the hill:
Time skiing down = distance ÷ speed skiing down
So, we have:
d/50 miles per hour
Since the total time is 0.367 hours, we can create the equation:
d/5 + d/50 = 0.367
To simplify the equation, let's find a common denominator:
(10d + d)/50 = 0.367
Simplifying further:
11d/50 = 0.367
Now, we can solve for "d":
11d = 0.367 × 50
11d = 18.35
d ≈ 1.668
Therefore, the distance from the bottom to the top of the hill is approximately 1.668 miles.
To find the distance from the bottom to the top of the hill, we need to calculate the total distance traveled during the round trip. Let's break down the problem into different parts:
Let's assume the distance from the bottom to the top of the hill is 'd' miles.
1. Calculate the time it takes to ride up the ski lift:
The man rides up the ski lift at a rate of 5 miles per hour. Since the distance is 'd', it will take him (d / 5) hours to reach the top.
2. Calculate the time it takes to ski back down the hill:
The man skis down the hill at a rate of 50 miles per hour. Since the distance is 'd', it will take him (d / 50) hours to reach the bottom.
3. Calculate the total time for the round trip:
The total time for the round trip is given as 22 minutes. However, the time for the ride up and the time for skiing down should add up to 22 minutes, as they are happening concurrently. We need to convert minutes to hours for consistency.
22 minutes is equivalent to (22/60) hours = 11/30 hours.
Since the time taken for the ride up and the time taken for skiing down should add up to 11/30 hours, we can create the following equation:
(d / 5) + (d / 50) = 11/30
Simplifying the equation:
(6d + d) / 50 = 11/30
(7d / 50) = 11/30
Cross-multiplying:
30 * 7d = 50 * 11
210d = 550
Now, we can isolate 'd' by dividing both sides of the equation by 210:
d = 550 / 210
d = 2.619 miles
Therefore, the distance from the bottom to the top of the hill is approximately 2.619 miles.