Please help me with the steps !i don't understand this.
The working arm of a tower crane is 192m high and has a length of 71.6 m. Suppose the hook reaches the ground and moves along the arm of the trolley,
a) determine max distance from hook to operator when trolley is fully extended at 71.6 m and min distance when trolley is closest to operator at 8.1 m.Hint: operator is at vertex of crane.
b) determine max and min angles of depression from the operator to the hook on the ground.
Sure, let's break down the problem step by step:
a) To determine the maximum distance from the hook to the operator when the trolley is fully extended at 71.6 m, we need to consider the length of the arm and the position of the operator.
1. Since the operator is at the vertex of the crane, it means the operator is at the top of the tower crane where the arm starts.
2. The total height of the arm of the tower crane is given as 192 m.
3. When the trolley is fully extended at 71.6 m, we can consider this as the length of the horizontal component of the crane arm.
4. To find the maximum distance from the hook to the operator, we need to use the Pythagorean theorem, which states that the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.
5. In this case, the hypotenuse represents the maximum distance from the hook to the operator, and the two sides are the height of the tower crane arm and the horizontal component of the arm (which is equal to the length of the fully extended trolley).
Using the Pythagorean theorem, we can calculate the maximum distance using the formula:
Maximum distance = √(Height of tower crane arm)^2 + (Length of fully extended trolley)^2
Substituting the given values:
Maximum distance = √(192^2 + 71.6^2)
b) To determine the minimum distance from the hook to the operator when the trolley is closest to the operator at 8.1 m, we can use a similar approach.
1. The minimum distance will occur when the trolley is closest to the operator, so we consider this as the length of the horizontal component of the arm.
2. Using the same Pythagorean theorem formula, we can calculate the minimum distance:
Minimum distance = √(Height of tower crane arm)^2 + (Length of trolley at closest position)^2
Substituting the given values:
Minimum distance = √(192^2 + 8.1^2)
b) To determine the maximum and minimum angles of depression from the operator to the hook on the ground, we can use trigonometry.
1. The angle of depression is the angle between a horizontal line and the line of sight (the line connecting the operator and the hook on the ground).
2. The maximum angle of depression occurs when the hook is at its maximum distance from the operator, and the minimum angle of depression occurs when the hook is at its minimum distance from the operator.
3. We can use the tangent function to calculate the angles of depression:
Maximum angle of depression = tan^(-1)(Height of tower crane arm / Length of fully extended trolley)
Minimum angle of depression = tan^(-1)(Height of tower crane arm / Length of trolley at closest position)
Using the given values, we can substitute them into the tangent function to calculate the angles.
(a) max: d^2 = 192^2 + 71.6^2
min: d^2 = 192^2 + 8.1^2
(a) max: tanθ = 192/8.1
min: tanθ = 192/71.6
Didja forget your Pythagorean Theorem and the definitions of the basic trig functions?