A ship is due south of a lighthouse. It sails on a bearing of 72* for 34 km when it is then due east of the lighthouse. Choose the one option which is closest to the distance (in km) of the ship from the lighthouse when it lies due east of the lighthouse.
Options
A 17.0 B 24.9 C 25.0 D 31.9 E 32.3 F 34.1
To solve this problem, we can use the concept of right-angled triangles and trigonometry.
Let's denote the distance of the ship from the lighthouse when it is due east as x km.
According to the problem, the ship initially sails south for 34 km. This forms the vertical side of the right-angled triangle.
Next, the ship sails on a bearing of 72 degrees for x km, which forms the horizontal side of the right-angled triangle.
Since the ship is due east of the lighthouse, the horizontal and vertical sides of the triangle must be equal in length.
Using trigonometry, we can find the value of x:
cos(72) = adjacent/hypotenuse
cos(72) = x/34
x = 34 * cos(72)
x ≈ 11.5 km
Therefore, the closest option to the distance of the ship from the lighthouse when it lies due east is option A: 17.0 km.
To find the distance between the ship and the lighthouse when they are due east of each other, we need to use trigonometry and the information given in the problem.
Let's assume the initial position of the ship to be point A, the final position (when it is due east of the lighthouse) to be point B, and the position of the lighthouse to be point L.
From the problem, we know that the ship sailed on a bearing of 72° for 34 km. This indicates that the ship traveled in a direction 72° clockwise from due north. We can imagine a right-angled triangle ABC, where AB represents the distance traveled by the ship and angle ABC is 72°.
To find the distance between the ship and the lighthouse (BL), we can use trigonometry (specifically, the sine function). We know that the sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse.
In this case, the length of the side opposite angle ABC is BL (the distance between the ship and the lighthouse), and the hypotenuse of the triangle is AB (the distance traveled by the ship). Therefore, we have:
sin(72°) = BL / 34
Rearranging the equation, we can solve for BL:
BL = 34 * sin(72°)
Using a scientific calculator to find the sine of 72° and then multiplying it by 34, we get:
BL ≈ 31.88
The closest option to this answer is D, 31.9 km.
Interesting correlation, you and S Rey spell Mathematics the same way.
Author is mathematician, not a navigator. The ship sailed on a HEADING of 72 degrees.
Anyway:
sin 72 = x/34
x = 34 (.95) = 32.3