A motorboat left a harbor and traveled to an island at an average rate of 18 knots. The average speed on the return trip was 10 knots. If the total trip took 14.0 hours, how far is the harbor from the island?
since time = distance/speed
d/18 + d/10 = 14
d = 90
5 hours out, 9 hours back = 14 total
v1 = an average velocity travel to island = 18 knopts
v2 = an average velocity on the return = 10 knots
t1 = time of travel to island
t2 = time on the return
L = Distance betwen harbor and island
t = t1 + t2 = total time of trip = 14 h
v1 = L / t1
18 = L / t1
L / t1 = 18
v2 = L / t2
10 = L / t2
L / t2 = 10
t = t1 + t2 = 14
t1 + t2 = 14 Subtract t1 to both sides
t1 + t2 - t1 = 14 - t1
t2 = 14 - t1
L / t2 = 10
L / ( 14 - t1 ) = 10
Now you must solve syatem of two equations:
L / t1 = 18
L / ( 14 - t1 ) = 10
The solutions are :
L = 90
t1 = 5
1 knot = 1 nautical mile per hour
L = 90 nautical mile
Proof :
v1 = L / t1 = 18 knots
v1 = 90 / 5 = 18 knots
t2 = 14 - t1
t2 = 14 - 5 = 9 h
v2 = L / t2 = 10 knots
v2 = 90 / 9 = 10 knots
To find the distance between the harbor and the island, we need to use the formula:
Distance = Rate x Time
Let's denote the distance as "d" and the time for the outbound trip as "t1".
The average speed on the outbound trip was 18 knots, so we can write:
18 knots = d / t1
Solving for t1:
t1 = d / 18 knots
Similarly, let's denote the time for the return trip as "t2".
The average speed on the return trip was 10 knots, so we can write:
10 knots = d / t2
Solving for t2:
t2 = d / 10 knots
We know that the total trip took 14.0 hours, so we have the equation:
t1 + t2 = 14.0 hours
Now, let's substitute the expressions for t1 and t2:
d / 18 knots + d / 10 knots = 14.0 hours
To solve this equation, we'll first find a common denominator:
(10 * d) / (18 knots * 10) + (18 * d) / (10 knots * 18) = 14.0 hours
Multiply both sides of the equation by (18 knots * 10):
10 * d + 18 * d = 14.0 hours * (18 knots * 10)
Simplifying:
28 * d = 2520 knots * hours
Now, divide both sides by 28:
d = 2520 knots * hours / 28
Simplifying:
d = 90 knots * hours / 1
Therefore, the distance between the harbor and the island is 90 nautical miles.
To find the distance between the harbor and the island, we can use the formula:
Distance = Rate * Time
Let's represent the distance from the harbor to the island as "d."
On the outbound trip, the motorboat traveled at an average rate of 18 knots for a certain amount of time, which we'll call "t". Therefore, the distance from the harbor to the island during the outbound trip is 18t.
On the return trip, the motorboat traveled at an average rate of 10 knots for the remaining time of the total trip (14.0 hours - t). So, the distance from the island back to the harbor is 10(14.0 - t).
Since the round trip covers the same distance, we can set up an equation:
18t = 10(14.0 - t)
Let's solve for t:
18t = 140 - 10t
28t = 140
t = 5
Now that we have the value of t, we can substitute it back into the equation to find the distance:
Distance = 18t
Distance = 18(5)
Distance = 90 knots
Therefore, the harbor is 90 knots away from the island.