To determine how far a boat is from shore, two radar stations 500 feet apart find the angles out to the boat, as shown in the given figure. Determine the distance of the boat from station 𝐴 and the distance of the boat from shore. Round your answers to the nearest whole foot.
Let’s call the position of the boat 𝐶. We can also add the distance between the radar stations, which is 500 feet. It’s also useful to calculate the third angle in this triangle. Since angles in a triangle add to 180 degrees, we can subtract 70 and 60 from 180 to find the measure of the angle at 𝐶. 180 minus 60 plus 70 is 50. So the measure of the angle at 𝐶 at the boat is 50 degrees.
Next, we’ll label the sides of our triangle. The side opposite station 𝐴 is lowercase 𝑎, the side opposite station 𝐵 is lowercase 𝑏, and the side opposite the boat, which we’ve called 𝐶, is lowercase 𝑐. Now, we have a non-right-angled triangle with two angles and an included side.
To calculate the distance of the boat from station 𝐴, which we’ve labelled as side 𝑏, we can use the law of sines. Remember we only need to use two parts of this. We know the length of side 𝑐 and we’re trying to find the length of the side marked 𝑏. So we’ll use 𝑏 divided by sin 𝐵 is equal to 𝑐 divided by sin 𝐶.
Substituting the values from our triangle then gives us 𝑏 divided by sin 60 is equal to 500 divided by sin 50. We can then solve this equation by multiplying both sides by sin 60. That gives us that 𝑏 is equal to 500 divided by sin 50 multiplied by sin 60 which is 565.257. Correct to the nearest whole foot, the distance from the boat to station 𝐴 is 565 feet.
Now, we need to calculate the distance of the boat from the shore. That’s this line, which is perpendicular to the shore. Now, we can see that we have a right-angled triangle, for which we know one length and one angle. We can use right angle trigonometry to work out the distance of the boat from the shore.
Let’s call that distance 𝑑. First, we’ll label our triangle. The hypotenuse is the longest side of the triangle. It’s opposite the right angle. The opposite is opposite the given angle and the adjacent is the other side of the triangle. It’s the one next to the given angle.
Since we know the length of the hypotenuse and we’re trying to calculate the length of the side opposite the angle, we need to use the sine ratio. sin 𝜃 is equal to the opposite divided by the hypotenuse. sine of 70 degrees is equal to 𝑑 divided by 565.25.
Notice how we’re using the exact value for the length of the side originally labelled 𝑏. This will prevent any errors from rounding too early. To solve our equation, we’ll multiply both sides by 565.25. That gives us that 𝑑 is equal to sine of 70 multiplied by 565.25, which is 531.168. That’s 531 feet correct to the nearest whole foot.