Kurt wants to sail his boat from a marina to an island 15 miles east of the marina. Along the course, there are several small islands they must avoid. He sails first on a heading of 70° and then on a heading of 120° (remember that headings angle measures rotated clockwise from the north). What is the total distance he travels before reaching the island?
To calculate the total distance Kurt travels, we need to break down the two legs of his journey separately.
First, let's find the distance Kurt travels on a heading of 70°. Since the island is 15 miles east of the marina, we can create a right-angled triangle where the leg adjacent to the 70° angle represents the distance traveled on this heading. We can use cosine to calculate this distance.
Using cosine:
cos(70°) = adjacent/hypotenuse
adjacent = cos(70°) x hypotenuse
Since the hypotenuse is the total distance to the island (15 miles), we can calculate the distance on a heading of 70° as:
distance_70 = cos(70°) x 15
Next, let's find the distance Kurt travels on a heading of 120°. Again, we create a right-angled triangle, this time with the leg adjacent to the 120° angle representing the distance traveled on this heading. Using cosine once more:
cos(120°) = adjacent/hypotenuse
adjacent = cos(120°) x hypotenuse
Since the hypotenuse is still the total distance to the island (15 miles), we can calculate the distance on a heading of 120° as:
distance_120 = cos(120°) x 15
Finally, to find the total distance he travels, we simply sum up these two distances:
total_distance = distance_70 + distance_120
By substituting the values and performing the calculations, we can determine the total distance Kurt travels before reaching the island.
To find the total distance Kurt travels before reaching the island, we need to break down his journey into two parts: sailing on a heading of 70° and sailing on a heading of 120°.
Part 1: Sailing on a heading of 70°
From the information given, we know that Kurt is initially sailing on a heading of 70°. Since he wants to reach the island located 15 miles east of the marina, we can use some trigonometry to determine the distance he travels on this heading.
Using the cosine function, we can determine the eastward distance traveled:
Distance = 15 miles * cos(70°)
Calculating this using a calculator, we find that the eastward distance traveled is approximately 5.05 miles.
Part 2: Sailing on a heading of 120°
After sailing on a heading of 70°, Kurt changes his heading to 120°. This heading is not directly towards the island, so we need to calculate the actual distance he travels on this heading using trigonometry.
Again, using the cosine function, we can determine the eastward distance traveled:
Distance = 15 miles * cos(120°)
Calculating this using a calculator, we find that the eastward distance traveled is approximately -7.5 miles. Here, the negative sign indicates that Kurt is traveling westwards.
Total Distance Traveled:
To find the total distance traveled, we add the distances traveled in each part:
Total Distance = Distance on heading of 70° + Distance on heading of 120°
= 5.05 miles + (-7.5 miles)
= -2.45 miles
The negative sign indicates that Kurt traveled 2.45 miles back towards the marina.
In conclusion, Kurt travels a total distance of approximately 2.45 miles before reaching the island.
If we draw a diagram and label the points in order M,A,B then
angle A is 130°
Now you can use the law of sines to find sides
m/sin20° = b/sin30° = 15/sin130°
The distance traveled is m+b