A 30-m tall flag pole and its shadow form the sides of a right triangle. On this particular day, the sun rises at 6 AM and sets at 6 PM. At what time of the morning (to the nearest minute) will the length of the shadow be exactly 40 m? (Hint: determine the angle of elevation of the sun)
H=Hight=30 m
L=Lenght of shadow
H/L=3/4=0.75
6AM+6PM=12 hours
12*(3/4)=36/4=9 hours
6AM+9hours=6AM+6hours+3hours=12AM+3hours=3PM
To determine the time of the morning when the length of the shadow is exactly 40 meters, we need to find the angle of elevation of the sun at that time.
Let's assume that the angle of elevation of the sun is θ.
In a right triangle, the tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side.
In this case, we can express the tangent of the angle of elevation as:
tan(θ) = height of the flagpole / length of the shadow
tan(θ) = 30 m / length of the shadow
Since we are given that the length of the shadow is 40 m, we can substitute this value into the equation:
tan(θ) = 30 m / 40 m
Now, we can solve for θ by taking the inverse tangent (arctan) of both sides of the equation:
θ = arctan(30 m / 40 m)
Using a calculator, you can find the inverse tangent of 30/40, which is approximately 36.87 degrees.
Next, we need to convert the angle from degrees to time.
In a 24-hour day, there are 360 degrees of rotation because the Earth completes one full rotation in 24 hours. Therefore, each degree corresponds to 4 minutes (60 minutes ÷ 360 degrees = 0.1667 minutes per degree).
To find the time in minutes, we can multiply the angle of elevation in degrees by the conversion factor:
Time (in minutes) = 36.87 degrees × 4 minutes/degree
Time (in minutes) ≈ 147.48 minutes
Rounded to the nearest minute, the length of the shadow will be exactly 40 meters at approximately 147 minutes after 6 AM.
To find the specific time, we need to add 147 minutes to 6 AM:
6 AM + 147 minutes = 8:27 AM