Draw the diagram. You can see that if the distance is x, then
20/x = tan30° = 1/√3
x = 20√3 m
20/x = tan30° = 1/√3
x = 20√3 m
First things first, we know that the tower is 20 meters high, and the angle of depression is 30 degrees. Now, picture this: the angle of depression is just like when you spot a slice of pizza on the floor and your mood goes "depressed" because it's out of reach. Anyway, back to the problem!
In this case, the angle of depression is the angle between the imaginary line from the top of the tower to the boat and the line parallel to the ground (think of it as a water slide). Now, let's call the distance from the front of the tower to the boat "x" meters.
Since we have this delightful right triangle formed, with the tower's height as one side and the distance to the boat as the hypotenuse, we can use some trigonometry. The tangent of an angle is the opposite side divided by the adjacent side. In this case, the tangent of 30 degrees can be written as:
tan(30 degrees) = 20 / x
Now, was that sailboat rented by a mathematician? Let's find out!
Using our trigonometric magic, we can rearrange the equation to solve for x:
x = 20 / tan(30 degrees)
Now, bring out your calculators (or your amazing mental math skills): tan(30 degrees) is approximately 0.577.
So, x = 20 / 0.577
Calculating that out, we find that the boat is approximately 34.64 meters away from the front of the tower.
Voila! The boat is floating around 34.64 meters away. Just don't forget to bring your sunscreen and your sense of humor with you on the journey!
Let's denote the distance from the front of the tower to the boat as "x" (in meters).
In this case, if we draw a right triangle with the tower height (20 meters) as the vertical side and "x" as the horizontal side, the angle of depression (30 degrees) would be the angle opposite the vertical side.
Now we can use the tangent function to relate the angle and the sides of the triangle:
tan(angle) = opposite / adjacent
tan(30°) = 20 / x
Now we need to solve for "x." We can rearrange the equation to isolate "x":
x = 20 / tan(30°)
Using a calculator, we can find the tangent of 30 degrees and calculate:
x = 20 / 0.57735
x = 34.64 meters (approximately)
Therefore, the boat is approximately 34.64 meters away from the front of the tower.