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 denote the distance from the harbor to the island as "d".
During the trip to the island, the motorboat's speed was 11 knots, and the time taken was "t1". So, the distance can be calculated as:
Distance (to the island) = 11 knots * t1
On the return trip, the motorboat's speed was 10 knots, and the time taken was "t2". The distance can be calculated as:
Distance (return trip) = 10 knots * t2
Since the total trip took 4.2 hours, we know that:
t1 + t2 = 4.2 hours
Since the distance to the island and the return trip distance are the same (as the motorboat is simply traveling back and forth), we can equate the distances:
11 knots * t1 = 10 knots * t2
Now, we can solve these equations to find the values of t1 and t2.
From the equation t1 + t2 = 4.2 hours, we can solve for one variable in terms of the other, and substitute it into the second equation.
Let's solve for t1 in terms of t2:
t1 = 4.2 - t2
Substituting this into the second equation:
11 knots * (4.2 - t2) = 10 knots * t2
Simplifying:
46.2 knots - 11 knots * t2 = 10 knots * t2
Combining like terms:
21 knots * t2 = 46.2 knots
Dividing both sides by 21 knots:
t2 = 46.2 knots / 21 knots
t2 ≈ 2.2 hours
Substituting this value back into the equation t1 = 4.2 - t2:
t1 = 4.2 - 2.2
t1 ≈ 2 hours
Now we can find the distance using the equation:
Distance = Speed * Time
Distance (to the island) = 11 knots * t1
Distance (to the island) ≈ 11 knots * 2 hours
Distance (to the island) ≈ 22 nautical miles
Therefore, the distance between the harbor and the island is approximately 22 nautical miles.
To find the distance between the harbor and the island, we need to use the formula: Distance = Speed x Time.
Let's denote the distance between the harbor and the island as 'd'.
On the outbound trip, the boat traveled from the harbor to the island at an average speed of 11 knots. Let's say it took 't1' hours for this trip.
So, according to the formula, the distance covered on the outbound trip is:
Distance1 = Speed1 x Time1
Distance1 = 11 knots x t1 hours
On the return trip, the boat traveled from the island back to the harbor at an average speed of 10 knots. Let's say it took 't2' hours for this trip.
So, the distance covered on the return trip is:
Distance2 = Speed2 x Time2
Distance2 = 10 knots x t2 hours
According to the given information, the total trip took 4.2 hours. This means that the sum of the outbound and return trip times is equal to 4.2 hours:
t1 + t2 = 4.2
Now, we can set up a system of equations to solve for 't1' and 't2':
Distance1 + Distance2 = d
11t1 + 10t2 = d
t1 + t2 = 4.2
To solve this system of equations, we'll use the method of substitution. First, we can solve the third equation for 't1' and substitute it into the second equation. Let's solve the third equation for 't1':
t1 = 4.2 - t2
Now, substitute this value of 't1' into the second equation:
11(4.2 - t2) + 10t2 = d
Now we have a single equation with 't2' as the only variable. We can solve this equation to find the value of 't2'.
11(4.2 - t2) + 10t2 = d
46.2 - 11t2 + 10t2 = d
46.2 - t2 = d
Since 'd' represents the distance between the harbor and the island, we can solve for 'd' using the equation:
d = 46.2 - t2
To find the value of 't2', we substitute the expression for 't1' from the third equation into the second equation and solve for 't2'. Once we have 't2', we can substitute it back into the equation for 'd' to find the distance between the harbor and the island.
distance to island --- n miles
x/11 + x/10 = 4.2
times 110
10x + 11x = 462
x = 22 nautical miles