Answer:
Step-by-step explanation:
I have drawn a figure to represent the given ship. See the attached image. This situation makes a triangle.
We are working to solve for x in this figure, how far the ship is from where it departed. To do this, we will create two right triangles. First, 180 degrees minus the 12 degrees means the interior angle at the turning point is 168°. This means that each of the two new angles is 84° because we bisect this angle to make our two right triangles.
Now, we can create an equation to find h. We will use the trigonometric functions of sine, cosine, and tangent to write equations. We are solving to find h, so that way we can use the Pythagorean theorem to find x.
[tex]\displaystyle cos\theta=\frac{adjacent}{hypotenuse}[/tex]
[tex]\displaystyle cos(84\°)=\frac{h}{40}[/tex]
[tex]\displaystyle 40cos(84\°)=h[/tex]
[tex]\displaystyle 4.1811\text{ miles}=h[/tex]
With this value of h, we can now find x. We will split x into two sections using the Pythagorean theorem, and then add them together.
a² + b² = c²
[tex]h^2 + (x_{1})^2 = 40^2[/tex]
[tex]4.1881^2 + (x_{1})^2 = 40^2[/tex]
[tex]17.54018 + (x_{1})^2 = 1,600[/tex]
[tex](x_{1})^2 = 1,582.45982[/tex]
[tex]x_{1} = \sqrt{1,582.45982} \approx 39.7801435[/tex]
a² + b² = c²
[tex]h^2 + (x_{2})^2 = 70^2[/tex]
[tex]4.1881^2 + (x_{2})^2 = 70^2[/tex]
[tex]17.54018 + (x_{2})^2 = 4,900[/tex]
[tex](x_{2})^2 =4,882.45982[/tex]
[tex]x_{2} =\sqrt{4,882.45982} \approx 69.8746[/tex]
x = 39.7801435 + 69.8746 = 109.6547 miles
Please note that your teacher may have rounded or not rounded at different points, or used different trigometric functions. As such, answers may differ. However, with the information given, my work should provide the correct answer. There are also many different ways to solve this question, this is the method I have used as I find it to be the easiest for me!
