Two Boats left the harbour at the same time. One travelled at 10km/h on a bearing of N47°E. The other travelled at 8km/h on a bearing of N79°E. How far apart were the boats after 78 mins? Round the distance to the nearest tenth of a kilometer. Use trigonometry.

To understand this you really need to sketch this out.

Think of the x/y axis on a graph, with the y-axis as North and the x-axis as East. Both boats are traveling from the North (top of the y-axis), toward East (down toward the x-axis).

The first bearing, forms a 47 deg. angle with the y-axis and the second bearing, goes a little further forming a 79 deg. angle with the y-axis.

The triangle formed, (I labeled ABC), with A at the origin, side c (between angle A and angle B), side b (between angle A and angle C) and side a (between angle B and angle C).

You need to find side a, the distance between the boats.

1 km/h / 60 min = 0.01667 km/min
First boat 10 km/h * 0.01667 = 0.1667 km/min * 78 min = 13 km
Second boat 8 km/h * 0.01667 = 0.1333 km/min * 78 min = 10.4 km

Angle A = 79 deg - 47 deg = 32 deg.
side c = 13 km (first boat's bearing)
side b = 10.4 km (second boat's bearing)

To find side a,
a^2 = b^2 + c^2 - 2bc cos A
a^2 = 10.4^2 + 13^2 - 2(10.4)(13)cos 32d
a^2 = 108.16 + 169 - 270.4 (.8)
a^2 = 277.16 - 216.32
a^2 = 60.8 km

To find the distance between the two boats after 78 minutes, we need to find the displacement of each boat and then calculate the distance between them using trigonometry.

First, let's convert the given time of 78 minutes to hours by dividing it by 60: 78 minutes ÷ 60 minutes/hour = 1.3 hours.

Now let's calculate the displacement for each boat using the formula:

Displacement = Speed × Time

For the first boat:
Displacement of the first boat = 10 km/h × 1.3 hours = 13 km.

For the second boat:
Displacement of the second boat = 8 km/h × 1.3 hours = 10.4 km.

Now we can use the Law of Cosines to find the distance between the two boats. The formula is:

Distance = √[(Displacement1)² + (Displacement2)² - 2(Displacement1)(Displacement2)cos(angle)]

The angle between the two boats can be calculated by finding the difference in their bearings:

Angle = angle1 - angle2 = 47° - 79° = -32°

Now let's substitute the values into the formula and calculate the distance:

Distance = √[(13)² + (10.4)² - 2(13)(10.4)cos(-32°)]

Using a calculator:

Distance ≈ √[169 + 108.16 - 270.4 × 0.848] ≈ √[277.16 - 229.6576] ≈ √47.5024 ≈ 6.9 km (rounded to the nearest tenth)

Therefore, the distance between the two boats after 78 minutes is approximately 6.9 kilometers.

To find the distance between the two boats after 78 minutes, we will first calculate the distance each boat has traveled during this time.

Boat 1 traveled at a speed of 10 km/h for 78 minutes, which is 1 hour and 18 minutes. Therefore, Boat 1 traveled a distance of 10 km/h * 1.3 hours = 13 km.

Boat 2 traveled at a speed of 8 km/h for 78 minutes, which is also 1 hour and 18 minutes. Thus, Boat 2 traveled a distance of 8 km/h * 1.3 hours = 10.4 km.

Now, we need to determine the distance between the two boats using trigonometry. We have the lengths of two sides of a triangle formed by the two boats and the distance between them as the unknown hypotenuse.

To find the angle between the two boats, we can subtract the two given bearings:

Angle = N47°E - N79°E

To get the angle in degrees, we need to convert the bearings to their equivalent angles. Bearing N47°E is equivalent to 47° and bearing N79°E is equivalent to 79°.

Angle = 79° - 47° = 32°

Now, we can use the cosine rule to find the distance between the two boats. The cosine rule states that in a triangle with sides a, b, and c, and the angle between sides a and b is C, the following equation holds:

c^2 = a^2 + b^2 - 2ab * cos(C)

Let's denote the distance between the two boats as c, the distance traveled by Boat 1 as a, and the distance traveled by Boat 2 as b.

Plugging in the values we calculated:

c^2 = 13^2 + 10.4^2 - 2 * 13 * 10.4 * cos(32°)

c^2 = 169 + 108.16 - 270.4 * cos(32°)

c^2 = 277.16 - 270.4 * cos(32°)

Now, we can calculate the distance c:

c = sqrt(277.16 - 270.4 * cos(32°))

Using a calculator, we can find that cos(32°) ≈ 0.8480. Substituting this value into the equation:

c = sqrt(277.16 - 270.4 * 0.8480)

c = sqrt(277.16 - 229.0032)

c = sqrt(48.1568)

c ≈ 6.93 km

Therefore, the boats are approximately 6.9 km apart after 78 minutes.