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 can use the formula:
Distance = Speed × Time
Let's assign variables to the given information:
Let x be the distance between the harbor and the island (in nautical miles).
Let t1 be the time taken from the harbor to the island.
Let t2 be the time taken from the island to the harbor.
Let v1 be the average speed from the harbor to the island (11 knots).
Let v2 be the average speed from the island to the harbor (10 knots).
Let the total trip time be T (4.2 hours).
Since the total trip time is the sum of the time to the island and the time back to the harbor:
T = t1 + t2
We also know that the time taken is equal to the distance divided by the speed:
t1 = x / v1
t2 = x / v2
Substituting these values back into the equation for the total trip time:
4.2 = x / 11 + x / 10
To solve this equation, we can simplify it by multiplying both sides by 110 (the least common multiple of 11 and 10):
4.2 * 110 = (x * 10) / 11 + (x * 11) / 10
462 = 10x / 11 + 11x / 10
Now, let's clear the fractions by finding a common denominator:
462 = (100x + 121x) / (11 * 10)
462 = (221x) / 110
Multiply both sides by 110 to isolate x:
462 * 110 = 221x
50,820 = 221x
Now, divide both sides by 221 to find the value of x:
50,820 / 221 ≈ 230.31
Therefore, the distance between the harbor and the island is approximately 230.31 nautical miles.