a motorboat left a harbor and traveled to an island at an average rate of 12 knots, the average speed on the return trip was 10 knots. If the total trip took 8.8 hours, how far is the harbor from the island?

let the distance be x nautical miles

x/12 + x/10 = 8.8
times 60
5x + 6x = 528
x = 48 n miles.

check:
one way = 48/12 = 4 hrs
other way = 48/10 = 4.8 , for a total of 8.8 hrs

To solve this problem, we can use the formula: Time = Distance / Speed.

Let's assume that the distance from the harbor to the island is "d" nautical miles.

On the outbound trip:
Time1 = d / 12 knots

On the return trip:
Time2 = d / 10 knots

The total time for the trip is given as 8.8 hours:
Time1 + Time2 = 8.8

Substituting the equations for Time1 and Time2:
d/12 + d/10 = 8.8

To simplify the equation, we need to find a common denominator for 12 and 10, which is 60:
(5d + 6d) / 60 = 8.8

Combining like terms:
11d / 60 = 8.8

We can cross-multiply to solve for "d":
11d = 8.8 * 60
11d = 528

Now, divide both sides by 11:
d = 528 / 11
d ≈ 48

Therefore, the distance from the harbor to the island is approximately 48 nautical miles.

To find the distance between the harbor and the island, we can use the formula:

Distance = Speed x Time

Let's calculate the time it took for the boat to reach the island:

Since the boat traveled at an average speed of 12 knots, we can divide the total trip time by the average speed to find the time taken for the outward journey:

Time for outward journey = Distance / Speed = Total trip time / Average speed = 8.8 hours / 12 knots

Calculating this, we get:

Time for outward journey = 0.7333 hours

Now, let's calculate the time it took for the boat to return to the harbor:

Since the boat traveled at an average speed of 10 knots, we can divide the total trip time by the average speed to find the time taken for the return journey:

Time for return journey = Distance / Speed = Total trip time / Average speed = 8.8 hours / 10 knots

Calculating this, we get:

Time for return journey = 0.88 hours

Now, we know that the time taken for the round trip is the sum of the time taken for the outward and return journeys:

Total trip time = Time for outward journey + Time for return journey

8.8 hours = 0.7333 hours + 0.88 hours

To find the distance between the harbor and the island, we can use the formula again:

Distance = Speed x Time

The distance of the round trip is the same as the distance between the harbor and the island. Therefore, we need to find the distance traveled in one direction only.

Let's calculate the distance traveled during the outward journey:

Distance for outward journey = Speed x Time = 12 knots x 0.7333 hours

Calculating this, we get:

Distance for outward journey = 8.8 nautical miles

Therefore, the distance between the harbor and the island is 8.8 nautical miles.