Moana (an expert sailor) is sailing from her home to a nearby island which is 215 miles north and 33 miles west. There is a constant ocean current of 0.50 knots moving from west to east. Moana can sail her boat at a cruising speed of 5.5 miles per hour in still water.

1. What angle should Moana sail to get to the island?

2. How long will it take her to get there?

3. Moana’s friend Maui is floating on a raft that is moving with the ocean
current (meaning, he is not traveling with respect to the water). Would
Maui see Moana’s boat traveling with a speed faster than, slower than, or
equal to what an observer standing on the shore would see? Explain your
reasoning.

1. To determine the angle Moana should sail to get to the island, we can use trigonometry. Let's consider the north and west directions as the positive x-axis and positive y-axis, respectively.

The north-south distance to the island is 215 miles, and the west-east distance is 33 miles. We can use the Pythagorean theorem to find the direct distance to the island:

distance^2 = (215 miles)^2 + (33 miles)^2
distance^2 = 46225 miles^2 + 1089 miles^2
distance^2 = 47314 miles^2

Taking the square root of both sides, we find the distance:

distance = √(47314 miles^2)
distance ≈ 217.47 miles

Now we can find the angle Moana should sail. Using trigonometric ratios, we can determine that:

tan(angle) = opposite/adjacent
tan(angle) = 33 miles/215 miles
angle = arctan(33 miles/215 miles)
angle ≈ 8.91 degrees

So, Moana should sail at an angle of approximately 8.91 degrees.

2. To calculate the time it will take for Moana to reach the island, we need to consider the speed of her boat in still water and the effect of the ocean current.

First, we need to determine the effective speed of Moana's boat due to the ocean current. The current is moving from west to east at a speed of 0.50 knots. Since 1 knot is equivalent to 1.15 miles per hour, the effective speed of the current is approximately 0.575 miles per hour (0.50 knots * 1.15 miles per hour).

To find Moana's effective speed, we subtract the speed of the current from her cruising speed in still water:

effective speed = cruising speed - current speed
effective speed = 5.5 miles per hour - 0.575 miles per hour
effective speed = 4.925 miles per hour

Now, we can calculate the time it will take for Moana to reach the island using the formula:

time = distance / effective speed
time = 217.47 miles / 4.925 miles per hour
time ≈ 44.18 hours

So, it will take Moana approximately 44.18 hours to reach the island.

3. Maui, who is floating on a raft with the ocean current, would observe Moana's boat moving with a speed slower than what an observer standing on the shore would see. This is because Maui's motion is already being carried by the ocean current. Since Maui and the ocean current are moving together, he would perceive Moana's boat moving relative to them.

However, an observer standing on the shore would see Moana's boat moving at its actual speed, which is the combination of its cruising speed and the current speed. Therefore, the observer on the shore would perceive Moana's boat moving faster than Maui would see it.

To determine the answers to these questions, we first need to analyze the situation and apply some principles of physics.

1. To find the angle Moana should sail, we can use the concept of vectors. The distance north (215 miles) and the distance west (33 miles) can be represented as two vectors. The resultant vector, which represents Moana's direction and distance relative to the island, can be found by adding these two vectors.

Using the Pythagorean theorem, we can calculate the magnitude of the resultant vector:
Magnitude = √(215^2 + 33^2) = √(46225 + 1089) = √47314 ≈ 217.45 miles

Next, we can find the angle by using trigonometry. The angle can be determined using the tangent function:
Angle = arctan(33/215) ≈ 8.75 degrees

Thus, Moana should sail at an angle of approximately 8.75 degrees.

2. To calculate the time it will take Moana to reach the island, we need to consider Moana's effective speed relative to the island. We'll break down her effective speed into two components: the speed in still water (5.5 miles per hour) and the effect of the ocean current (0.50 knots moving from west to east).
To calculate the effective speed, we can use the concept of vector addition again.
The eastward current won't contribute to Moana's progress since she is moving perpendicular to it. Therefore, we need to consider the northward current.

To convert the current speed from knots to miles per hour: 0.50 knots * 1.15078 miles per hour (conversion factor) = 0.57539 miles per hour.

The effective speed in the northward direction is:
Effective speed = Cruising speed in still water + Northward current speed
Effective speed = 5.5 + 0.57539 ≈ 6.08 miles per hour

Now, we can calculate the time using the formula: Time = Distance / Speed
Time = 217.45 miles / 6.08 miles per hour ≈ 35.7 hours

Therefore, it will take Moana approximately 35.7 hours to reach the island.

3. Maui, who is floating on a raft, is not traveling with respect to the water. From his perspective, the water current appears to be at rest.
An observer standing on the shore will see both Moana's boat and the ocean current. Since the observer on the shore is stationary, they will perceive Moana's boat as moving with the sum of Moana's boat speed and the ocean current speed. Therefore, an observer on the shore will see Moana's boat traveling faster than Maui sees it.

In summary, Moana should sail at an angle of approximately 8.75 degrees, it will take her approximately 35.7 hours to reach the island, and an observer standing on the shore will see Moana's boat traveling faster than Maui sees it.

now go back and get t

faster, headed toward

Sail At angle T west of North

speed North = 5.5 cos T
speed west = 5.5 sin T- 0.5
now sail for t hours
speed north * t = 215
speed west *t = 33
so
(5.5 cos T) t = 215
(5.5 sin T- 0.5) t = 33
so
t = 39 / cos T
(5.5 sin T- 0.5) 39 / cos T = 33
5.5 sin T - 0.5 = (33/39) cos T = 0.846 cos T
5.5 sin T - .846 cos T - 0.5 = 0
5.5 sqrt (1-cos^2 T)= .846 cos T + 0.5
sqrt (1 - cos^2 T) = 0.154 cos T + .091
1 - cos^2 T =.0237 cos^2 T + .0474 cos T + .00828
0 = 1.0237 cos^2 T + .0474 cos T - 0.992
solve quadratic
https://www.mathsisfun.com/quadratic-equation-solver.html
cos T = .962
so
T = 15.85
so sail 15.85 deg west of north