A motorboat left a harbor and traveled to an island at an average rate of 11 knots. The average speed on the return trip was 10 knots. If the total trip took 4.2 hours, how far is the harbor from the island
To find the distance between the harbor and the island, we need to use the formula:
Distance = Speed * Time
Let's assume that the distance between the harbor and the island is 'D' nautical miles.
First, we calculate the time it took to reach the island. The average speed on the way to the island was 11 knots, and let's say it took 't1' hours to reach the island.
So, the distance traveled from the harbor to the island is:
Distance = Speed * Time
D = 11 * t1
Next, we calculate the time it took to return from the island to the harbor. The average speed on the return trip was 10 knots, and let's say it took 't2' hours to return.
The distance traveled from the island to the harbor is the same as the distance from the harbor to the island, so:
Distance = Speed * Time
D = 10 * t2
Since the total trip took 4.2 hours, we can express this in terms of the two times:
Total Time = Time to reach the island + Time to return
4.2 = t1 + t2
We now have a system of equations:
D = 11 * t1
D = 10 * t2
4.2 = t1 + t2
To solve this system, we'll use substitution. From the first equation, we get:
t1 = D / 11
Now substitute t1 in the third equation:
4.2 = D / 11 + t2
Simplifying this equation gives:
4.2 = D / 11 + t2
4.2 = D / 11 + D / 10
To solve for D, let's find a common denominator:
4.2 = (10D + 11D) / (11 * 10)
4.2 = 21D / 110
4.2 * 110 = 21D
D = (4.2 * 110) / 21
D ≈ 22 nautical miles
Therefore, the harbor is approximately 22 nautical miles from the island.