Kurt wants to sail his boat from a marina to an islnd 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 degrees and then on a heading of 120 degrees. What is the total distance he travels before reaching the island?
70 is 20 north of east
120 is 30 south of east
draw that triangle with your two courses and the x (east) axis
your triangle has angles 20, 30 with leg 15 between
the angle at the top, opposite the 15 miles is therefore
180 - 50 = 130
then law of sines
sin30/leg1 = sin130/15
so leg1 = 9.8
sin20/leg2 = sin130/15
so leg2 = 6.7
9.8+6.7 = 16.5 miles
To find the total distance Kurt traveled before reaching the island, we need to calculate the length of each leg of his journey.
First, let's consider the leg where Kurt sails on a heading of 70 degrees. This leg can be represented as a right-angled triangle, where the side opposite the 70-degree angle is the distance he traveled (let's call it leg1). Since we know the distance east of the marina is 15 miles, we can use the trigonometric function cosine to calculate the length of the leg1:
cos(70 degrees) = adjacent leg / hypotenuse
cos(70 degrees) = leg1 / 15 miles
We can rearrange the equation to solve for leg1:
leg1 = cos(70 degrees) * 15 miles
Now, let's consider the leg where Kurt sails on a heading of 120 degrees. Similarly, this leg can also be represented as a right-angled triangle. The side opposite the 120-degree angle is the distance he traveled (let's call it leg2). To find the length of leg2, we can again use the cosine function:
cos(120 degrees) = adjacent leg / hypotenuse
cos(120 degrees) = leg2 / 15 miles
Solving for leg2:
leg2 = cos(120 degrees) * 15 miles
Finally, to find the total distance Kurt traveled, we sum up the lengths of leg1 and leg2:
Total distance = leg1 + leg2
Now, let's perform the calculations:
leg1 = cos(70 degrees) * 15 miles
leg1 = 0.3420 * 15 miles
leg1 = 5.13 miles
leg2 = cos(120 degrees) * 15 miles
leg2 = -0.5 * 15 miles
leg2 = -7.5 miles (negative value indicates sailing in the opposite direction)
Total distance = leg1 + leg2
Total distance = 5.13 miles + (-7.5 miles)
Total distance = -2.37 miles (negative value indicates sailing in the opposite direction)
Therefore, Kurt traveled a total distance of approximately 2.37 miles in the opposite direction before reaching the island.