The light on the Prince Shoal lighthouse is 25m above water level. From a position behind the light, the angle of the depression of the sailboat is 12 degrees. How far is the sailboat from the lighthouse, to the nearest metre?
(I have no idea how the diagram would look - it uses the primary trigonometry ratios to solve for the distance)
this diagram is no trick.
Draw a horizontal line: the water
draw a vertical line: the lighthouse.
Where the vertical line hits the water, call that point B: bottom of tower
At the top, label it T.
The height of the vertical line is 25
Now somewhere out on the water, label a point S: sailboat.
The line TS makes a 12° angle with the water. (No, it doesn't have to really be 12° -- just label it that way.
Now, the distance BS (from the boat to the base of the lighthouse) can be found using
25/BS = tan 12°
now, was that so hard? All you have to do is take one piece of information at a time, and use it as described.
To solve this problem, we can use trigonometry, specifically the tangent function. The tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.
Let's start by drawing a diagram to visualize the problem. Imagine a right triangle with the sailboat as the adjacent side, the distance from the base of the lighthouse to the sailboat as the opposite side, and the line from the top of the lighthouse to the sailboat as the hypotenuse.
Now, we know that the angle of depression from behind the lighthouse is 12 degrees, which means the angle between the line from the top of the lighthouse to the sailboat and the horizontal line is 12 degrees.
Next, we need to find the length of the opposite side, which is the distance from the base of the lighthouse to the sailboat. To do that, we can use the tangent function:
tan(angle) = opposite / adjacent
In this case, the opposite side is the distance from the base of the lighthouse to the sailboat, and the adjacent side is the vertical height of the lighthouse. We know that the height of the lighthouse is 25m.
So, we can write the equation as:
tan(12 degrees) = opposite / 25m
Now, we can solve for the opposite side:
opposite = tan(12 degrees) * 25m
Using a calculator, we can find that the tangent of 12 degrees is approximately 0.2126. Multiplying this by 25m, we get:
opposite ≈ 0.2126 * 25m ≈ 5.315m
Therefore, the distance from the sailboat to the lighthouse, to the nearest meter, is approximately 5 meters.