Sara needs to needs to trek from an oasis to a destination 10 miles away across a barren desert. However, she finds herself dealing with the following limitations:
a) Crossing one mile of desert requires using 1 gallon of water.
b) She can only carry 6 gallons of water at a time from the water-abundant oasis where she begins.
c) She can drop a water cache (of any amount of water from the supply she is carrying at that moment) at any of the nine stops along the route, and then pick up any part of the cache on a later trip.
The question is: What's the minimum number of times Sara must leave the oasis in order to cross the entire 10 mile span of desert?
It appears she needs to make 4 trips to the oasis.
To determine the minimum number of times Sara must leave the oasis in order to cross the entire 10-mile span of desert, we need to consider the following factors:
1. Crossing one mile of desert requires using 1 gallon of water.
2. She can only carry 6 gallons of water at a time.
3. She can drop water caches at any of the nine stops along the route and pick them up later.
Given these limitations, let's break down the trek into smaller sections:
Section 1: Oasis to Stop 1 (1 mile)
Sara needs to carry 1 gallon of water to cross this section.
Section 2: Stop 1 to Stop 2 (1 mile)
Sara needs to carry another 1 gallon of water to cross this section.
Section 3: Stop 2 to Stop 3 (1 mile)
It would be best if Sara picks up the gallon of water that she dropped at Stop 1 to use for this section. So, she does not need to carry any additional water.
Section 4: Stop 3 to Stop 4 (1 mile)
Sara needs to carry 1 gallon of water to cross this section.
Section 5: Stop 4 to Stop 5 (1 mile)
Sara can pick up the gallon of water she dropped at Stop 3, so she does not need to carry any additional water.
Section 6: Stop 5 to Stop 6 (1 mile)
Sara needs to carry 1 gallon of water to cross this section.
Section 7: Stop 6 to Stop 7 (1 mile)
Sara can pick up the gallon of water she dropped at Stop 5, so she does not need to carry any additional water.
Section 8: Stop 7 to Stop 8 (1 mile)
Sara needs to carry 1 gallon of water to cross this section.
Section 9: Stop 8 to Stop 9 (1 mile)
Sara can pick up the gallon of water she dropped at Stop 7, so she does not need to carry any additional water.
Section 10: Stop 9 to Destination (1 mile)
Sara needs to carry 1 gallon of water to cross this section.
Considering all these sections, Sara needs to leave the oasis 6 times in order to cross the entire 10-mile span of desert: once for each of the 6 sections where she needs to carry 1 gallon of water.
Note: It is assumed that Sara will only drop water caches when necessary and pick them up later, minimizing the number of times she needs to carry additional water.
To determine the minimum number of times Sara needs to leave the oasis to cross the entire 10-mile span of desert, we need to consider the distance she can cover with the amount of water she can carry and the distance she can cover for each gallon of water.
Given that crossing one mile of desert requires using 1 gallon of water, this means that Sara can only travel for a maximum of 6 miles with the 6 gallons of water she can carry from the oasis.
Therefore, the first calculation is to determine how many trips Sara needs to make with her 6-gallon water supply.
Total number of trips = Total distance / Distance covered per trip using 6 gallons
In this case, the total distance is 10 miles, and each trip covers a distance of 6 miles using 6 gallons of water.
Total number of trips = 10 miles / 6 miles = 1.66 trips
Since Sara cannot make a fraction of a trip, we round up the number of trips to the nearest whole number.
Total number of trips = 2 trips
Therefore, Sara needs to make a minimum of 2 trips from the oasis to cross the entire 10-mile span of desert.
During the first trip, she will carry 6 gallons of water, consume 6 gallons to cross 6 miles, and drop a water cache at the end of the 6 miles. Then, on the second trip, she will carry the remaining 4 gallons of water (6 gallons - 2 gallons used on the first trip), consume 4 gallons to cross the remaining 4 miles, and reach her destination.
The strategy of dropping water caches and picking them up on later trips allows Sara to make the journey with a limited water supply.