A bamboo pole 20 ft. long is erected vertically. From point 50 ft. from its base, what is the angle of elevation of the top?
it is the angle z such that
tan z = 20/50
To find the angle of elevation of the top of the bamboo pole, we can use trigonometry.
Let's call the angle of elevation θ.
From the given information, we have a right triangle formed by the bamboo pole, the ground, and the line connecting the point 50 ft from the base to the top of the bamboo pole.
Using trigonometry, the tangent function can help us find the angle:
tan(θ) = opposite / adjacent
In this case, the opposite side is the height of the bamboo pole, which is 20 ft, and the adjacent side is the distance from the point 50 ft from the base to the pole.
Let's find the adjacent side of the triangle using the Pythagorean theorem:
adjacent^2 + opposite^2 = hypotenuse^2
(50 ft)^2 + (20 ft)^2 = hypotenuse^2
2500 ft^2 + 400 ft^2 = hypotenuse^2
2900 ft^2 = hypotenuse^2
hypotenuse = √(2900 ft^2) = 53.85 ft (rounded to two decimal places)
Now that we know the adjacent side is 53.85 ft, we can find the angle of elevation using the tangent function:
tan(θ) = 20 ft / 53.85 ft
Using a calculator or a trigonometric table, we take the arctangent (inverse tangent) to find the angle:
θ = arctan(20 ft / 53.85 ft)
Using a calculator, we find:
θ ≈ 20.02 degrees (rounded to two decimal places)
Therefore, the angle of elevation of the top of the bamboo pole is approximately 20.02 degrees.
To find the angle of elevation of the top of the bamboo pole, you can use trigonometry. Specifically, you can use the tangent function.
The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side. In this case, the angle of elevation is the angle between the ground and the line of sight from the point 50 ft. from the base to the top of the pole.
Let's label the height of the pole as "h" (since we don't know it yet) and the distance from the point 50 ft. from the base to the base of the pole as "d" (which is also 50 ft. in this case).
Using the tangent function, we can write:
tan(angle) = h / d
We want to solve for the angle, so we rearrange the formula:
angle = arctan(h / d)
Now we can plug in the known values:
angle = arctan(h / 50)
Next, we need to determine the value of "h". Since the pole is vertical, the height is equal to the length of the pole, which is given as 20 ft.
Now we can substitute this value into the equation:
angle = arctan(20 / 50)
Simplifying further:
angle = arctan(2/5)
Using a scientific calculator or a trigonometric table, you can find that arctan(2/5) is approximately 21.8 degrees.
Therefore, the angle of elevation of the top of the bamboo pole is approximately 21.8 degrees.