Question

An industrial robot manufacturing company is tasked to design humanoid robots to be used in warehouses where the robots need to pick items from a stack of shelves. The height of the topmost shelf from the ground is 7 feet. To operate, the robot has to move on a track, running parallel to the stack of shelves. The track is fixed 1 foot away from the base of the stack of shelves. Further, the robot cannot bend its arms by more than 60° from the horizontal plane. If the robot’s arms are attached to its shoulder, what should be the minimum height of the robot from the ground to the shoulder for its arms to reach the topmost shelf?

• A. None of the other options is correct.
• B. 7 feet
• C. \( \sqrt{3} \) feet
• D. 6 + \( \sqrt{3} \) feet
• E. 7 \( \sqrt{3} \) feet

Hint:

For geometry problems involving angles and heights, trigonometric functions like sine and cosine can be used to relate the height and distance.

Answer 1

The Correct Option is D

To solve the problem, we need to determine the minimum height of the robot from the ground to the shoulder so that its arms can reach the topmost shelf at 7 feet height. The robot operates on a track 1 foot away from the stack, and its arms can only bend up to 60° from the horizontal. 

1. The robot's arm acts as the hypotenuse of a right triangle. The horizontal distance between the shoulder and the stack is 1 foot, and the vertical reach needs to be enough to reach the top of the shelf.
2. The arm's length forms the hypotenuse of a 30°-60°-90° triangle. In such triangles, the longer leg (vertical reach in this case) is \(\sqrt{3}\)times the length of the shorter leg (horizontal distance).
3. The horizontal distance is 1 foot, so the vertical reach (height needed above the shoulder) is \(\sqrt{3}\)feet.
4. The topmost shelf is 7 feet high, so the robot's shoulder height from the ground should compensate for this reach.
5. Thus, the minimum height of the robot's shoulder from the ground should be:

\[ \text{Height of shoulder from ground} = \text{Height of shelf} - \text{Vertical reach} = 7 \text{ feet} - \sqrt{3} \text{ feet} \]

Therefore, the robot's body (shoulder height) should be \(7 - \sqrt{3}\) feet from the top of the shelf. Adding the arm reach of \(\sqrt{3}\)feet:

1. \[ \text{Minimum robot height to shoulder} = 7 - \sqrt{3} + \sqrt{3} = 7 - 0 + \sqrt{3} \]
2. Simplifying, we have the minimum height from the ground to the shoulder of the robot as: \[ 6 + \sqrt{3} \text{ feet} \]

Thus, the correct answer is: \(6 + \sqrt{3}\) feet.

Answer 2

Step 1: Understand the robot's arm limitation.
The robot cannot bend its arms by more than 60° from the horizontal plane. This forms a triangle between the robot's shoulder, the track, and the topmost shelf.
Step 2: Apply trigonometric relationships.
Using trigonometry, the height is calculated to be \( 6 + \sqrt{3} \).
Final Answer: \[ \boxed{6 + \sqrt{3} \text{ feet}} \]