A hinged crane which measures 19m makes an angle of 50° with the ground. A malfunction causes the lock on the hinge to fail and the top part (of 14.6m) of the crane swings down. How far from the base of the crane does the top hot the ground.
Draw a diagram. The 14.6m part and the 4.4m part form two sides of a triangle. You want the 3rd side. Using the law of sines,
14.6/sin50° = 4.6/sinB
now you have one more angle, so the 3rd angle C is easy to find, and the side you want is now easy.
c/sinC = 14.6/sin50°
To find the distance from the base of the crane to where the top hits the ground, we need to use trigonometry.
Let's call the distance we want to find "x".
We know that the crane measures 19m in total and makes an angle of 50° with the ground. Therefore, the vertical component (height) of the crane can be found using the sine function:
sin(50°) = height / 19m
Rearranging the equation, we get:
height = 19m * sin(50°)
height ≈ 14.52m
Since the top part of the crane is 14.6m long, the distance from the base to where it hits the ground (x) can be found by subtracting the height of the crane from the length of the top part:
x = 14.6m - 14.52m
x ≈ 0.08m
Therefore, the top part of the crane hits the ground approximately 0.08 meters away from the base.
To find out how far from the base of the crane the top hits the ground, we can use trigonometry.
Let's start by drawing a diagram to visualize the situation. We have a right triangle, where the height of the crane is 19m, the length of the top part is 14.6m, and the angle between the ground and the crane is 50°.
Now, to find the distance from the base of the crane to the point where the top hits the ground, we can use the sine function. The sine function relates the opposite side of a right triangle to the hypotenuse. In this case, the opposite side is the distance from the base to the point where the top hits the ground, and the hypotenuse is the length of the crane.
The sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. In mathematical terms, we can write this as:
sin(angle) = opposite/hypotenuse
Substituting the known values:
sin(50°) = opposite/19m
Now, rearranging the equation to solve for the opposite side, we have:
opposite = sin(50°) * 19m
Using a scientific calculator, we can find the sine of 50°, which is approximately 0.766. Multiplying this by 19m gives us:
opposite = 0.766 * 19m
opposite ≈ 14.554m
Therefore, the distance from the base of the crane to the point where the top hits the ground is approximately 14.554m.