A small island is 2 km off shore in a large lake .A woman on the island can row her boat 10 km/hr and can run at a speed of 20 km/hr. If she rows to the closest point of the straight shore, she will land 6 km from a village on the shore. Where should she land to reach the village most quickly by a combination of rowing and running?
Well, it seems like this woman is a real lover of both rowing and running. Personally, I'd suggest she invent a new sport called "row-running" and become famous, but that might not help her reach the village quickly!
To figure out the best strategy for her, let's do some calculations. If the woman rows directly to the closest point on the shore, she'll land 6 km from the village. Then, she would have to run the remaining distance of 6 km.
Alternatively, she could row in a diagonal direction towards a point on the shore that is closer to the village, let's call it point X. By applying some math magic, we can calculate that the rowing distance to point X would be approximately 4.24 km.
Now, if she lands at point X, the remaining distance she would have to run to reach the village would be approximately 4 km.
So, by choosing to row towards point X, the woman only has to run 4 km instead of 6 km. And since her running speed is twice her rowing speed, it would be faster for her to row towards point X.
Remember, it's important to stay positive during these kinds of activities. Who knows, maybe she'll find a hidden treasure on her way to the village and become a wealthy row-running explorer!
To determine the most efficient way for the woman to reach the village, we need to analyze two scenarios: rowing directly to the village from the closest point on the shore or rowing to a different point on the shore where the woman can switch to running.
First, let's calculate the time it takes for the woman to reach the closest point on the shore by rowing directly. The distance between the island and the closest point on the shore is 2 km, and her rowing speed is 10 km/hr. Therefore, it would take her 2 km / 10 km/hr = 0.2 hours (or 12 minutes) to reach the closest point on the shore.
Next, let's calculate the time it takes for the woman to reach the village by rowing to a different point on the shore and then running. The distance between the island and the new landing point is still 2 km. However, to calculate the time it takes for her to reach the village, we need to consider the total distance she would have to run.
Since she lands 6 km from the village, the total distance she needs to run is 6 km. Her running speed is 20 km/hr. Therefore, it would take her 6 km / 20 km/hr = 0.3 hours (or 18 minutes) to run from the new landing point to the village.
Adding the time it takes for rowing and running together, we have:
0.2 hours (row to the closest point on the shore) + 0.3 hours (run to the village) = 0.5 hours (or 30 minutes).
Comparing the two scenarios, it takes the same amount of time for the woman to reach the village whether she rows directly to the closest point on the shore or rows to a different point and then runs to the village. Therefore, it doesn't matter where she lands; the total time to reach the village will be the same.
To find the optimal landing point for the woman to reach the village most quickly, we need to consider the combination of rowing and running speeds.
Let's denote the distance from the island to the village as "d." If the woman rows directly towards the village, she covers (d + 2) km by rowing and d km by running.
The time it takes to row a distance of (d + 2) km is given by (d + 2)/10, as her rowing speed is 10 km/hr. Similarly, the time it takes to run a distance of d km is d/20, as her running speed is 20 km/hr.
The total time required to reach the village is the sum of these two times, so we can express the total time as:
Time = (d + 2)/10 + d/20
To minimize the time, we can differentiate this expression with respect to d and find the value of d that makes the derivative zero:
d(Time)/d = (1/10) - (1/20) = 1/20
Simplifying, we have:
1/20 = (d + 2)/10 + d/20
1/20 = (2d + d + 2)/20
1 = 2d + d + 2
2d + d = 1 - 2
3d = -1
d = -1/3
However, since negative distances do not make sense, we can discard this solution.
Therefore, there is no minimum distance for the woman to land exactly on the village. The woman should instead land at the closest point to the village, which is 6 km from the village.
So, to reach the village most quickly, the woman should row to a point on the shore that is 6 km away from the village and then run the remaining distance to the village.
row must be sqr rt of (x^2+4); therefore the answer is 2 sqr rt of 3 allover 3 km
if she lands a distance x from the closest point on shore, she is 6-x from the village. So, she will
row √(x^2+36) km
run 6-x km
So, her travel time will be
t = √(x^2+36)/10 + (6-x)/20
dt/dx = (2x - √(x^2+36)) / 20√(x^2+36)
Since the denominator is never zero, dt/dx=0 when
2x - √(x^2+36) = 0
x = 2√3
So, she needs to land 2√3 km from the nearest point on shore.