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?
To determine where the woman should land to reach the village most quickly, we need to analyze the time it takes for her to row and run to the village from different landing points on the shore.
Let's assume the woman rows to a point A on the shore, which is 6 km away from the village. The distance from the island to point A would be the hypotenuse of a right-angled triangle with sides 2 km (distance from the island to the shore) and 6 km (distance from point A to the village).
Using the Pythagorean theorem, we can calculate the distance from the island to point A:
(distance from the island to point A)^2 = (distance from the island to the shore)^2 + (distance from point A to the village)^2
(distance from the island to point A)^2 = 2^2 + 6^2
(distance from the island to point A)^2 = 4 + 36
(distance from the island to point A)^2 = 40
(distance from the island to point A) = √40 ≈ 6.3246 km
Now, let's calculate the time it takes for the woman to row from the island to point A. The time can be calculated using the formula:
Time = Distance / Speed
Time taken to row from the island to point A = (distance from the island to point A) / (rowing speed)
Time taken to row from the island to point A = 6.3246 km / 10 km/hr
Time taken to row from the island to point A ≈ 0.6325 hr (approximately)
Next, let's calculate the time it takes for the woman to run from point A to the village, which is 6 km away. We can calculate the time using the formula:
Time = Distance / Speed
Time taken to run from point A to the village = (distance from point A to the village) / (running speed)
Time taken to run from point A to the village = 6 km / 20 km/hr
Time taken to run from point A to the village = 0.3 hr (approximately)
Now, let's calculate the total time for the woman to reach the village by rowing to point A and then running to the village:
Total time = Time taken to row from the island to point A + Time taken to run from point A to the village
Total time = 0.6325 hr + 0.3 hr
Total time ≈ 0.9325 hr (approximately)
By performing the same calculations for various landing points on the shore, it can be determined which landing point results in the shortest total time.