Draw a sketch in each question leave the answer in surd form with rational denominator
From the top of a tower, the angle of depression of a boat is 30degree. If the tower is 20m high how far is the boat from the foot of the tower
if you drew the sketch, and reviewed your basic trig functions, you can see that
20/x = tan30°
Make a sketch and using the basic trig ratios, which you MUST learn, .....
x/20 = tan30° , where x is the distance between the boat and the tower along the horizontal.
= 20tan30° = .....
oops, read the angle wrong,
use oobleck's solution
To find the distance between the boat and the foot of the tower, we can use the trigonometric concept of the tangent function.
First, let's draw a sketch to visualize the situation:
T
|\
| \
| \
| \
| \
B X
|__________|
In the sketch, T represents the top of the tower (of height 20m), B represents the boat, and X represents the foot of the tower. The angle of depression from the top of the tower to the boat is given as 30 degrees.
Now, let's apply the tangent function:
tan(angle) = opposite / adjacent
In this case, the opposite side is the height of the tower (20m) and the adjacent side is the distance between the boat and the foot of the tower (BX, which we need to find).
So, we can write the equation as:
tan(30°) = 20m / BX
Now, we can solve for BX:
BX = 20m / tan(30°)
To calculate this value, we need to convert the angle from degrees to radians because trigonometric functions typically operate in radians. The conversion is as follows:
angle in radians = (angle in degrees * π) / 180°
So, in radians, the angle of depression is:
30° * π / 180° = π / 6
Now, we can substitute this value into the equation:
BX = 20m / tan(π / 6)
Using a scientific calculator, evaluate the tangent of π / 6: tan(π / 6) = √3 / 3
Substitute this value back into the equation:
BX = 20m / (√3 / 3)
Since we want the answer in surd form with a rational denominator, we rationalize the denominator by multiplying both the numerator and denominator by √3:
BX = (20m / (√3 / 3)) * (√3 / √3)
= (20m * √3) / 3
Therefore, the distance between the boat and the foot of the tower is (20√3 / 3) meters.