with its arm fully extended to a length of 16ft, the maximum height that a man lift can reach is 18ft. The lowest height is 4ft, what is the position where the arm of the lift is horizontal.
a) if a man gets on the lift and the arm is lifted through an angle 15degrees,what would be the final height of the man, to the nearest tenth of a foot?
b) to the nearest degrees, what is the maximum angle that can be formed between the arm in its lowest position and the arm in its highest position.
c)as a man is lifted, he travels both vertically and horizontally as the arm moves. how far will a man travel horizontally between the lowest position and the highest position, to the nearest tenth of a foot.
To solve these questions, we can use trigonometry and the concept of right-angled triangles.
To answer part a:
The man lift is fully extended to a length of 16ft (hypotenuse) and has a maximum height of 18ft (opposite side).
Let's assume the arm position where it is horizontal is angle A. In this case, the side opposite to angle A is the maximum height of 18ft, and the hypotenuse is 16ft.
Using the sine function, we can find the value of angle A:
sin(A) = opposite/hypotenuse
sin(A) = 18/16
A = sin^(-1)(18/16)
A ≈ 69.3 degrees
Now, if the arm is lifted through an angle of 15 degrees, we need to find the new height. Let's call this new angle B.
Using the sine function again:
sin(B) = height/16
sin(15) = height/16
height = sin(15) * 16
height ≈ 4.1ft (to the nearest tenth of a foot)
So, the final height of the man would be approximately 4.1ft.
To answer part b:
To find the maximum angle between the arm in its lowest position (4ft) and its highest position (18ft), we can use the inverse sine function:
max angle = sin^(-1)(18/16)
max angle ≈ 69.3 degrees
Therefore, the maximum angle that can be formed between the arm in its lowest position and its highest position is approximately 69.3 degrees.
To answer part c:
As the arm moves, the man travels vertically and horizontally. To find the horizontal distance traveled, we can use the cosine function:
cos(A) = base/hypotenuse
cos(69.3) = base/16
base = cos(69.3) * 16
base ≈ 7.5ft (to the nearest tenth of a foot)
Therefore, the man will travel approximately 7.5ft horizontally between the lowest and highest positions.