Questions: A robot begins at point C and travels in a straight line across a warehouse floor to point A where it picks up some merchandise. It then turns a 37.0° corner and travels 46.8 m to point B where it drops off the merchandise, as in the figure to the right. If the robot must now turn a 28.0° corner to return to its original position, how far must it travel to get there? Describe how to solve this problem. To determine the desired distance, use the given information to determine and then use the formula The robot must travel to travel to its original position. (Round to one decimal place as needed.)

A robot begins at point C and travels in a straight line across a warehouse floor to point A where it picks up some merchandise. It then turns a 37.0° corner and travels 46.8 m to point B where it drops off the merchandise, as in the figure to the right. If the robot must now turn a 28.0° corner to return to its original position, how far must it travel to get there?

Describe how to solve this problem.
To determine the desired distance, use the given information to determine and then use the formula
The robot must travel to travel to its original position.
(Round to one decimal place as needed.)

Transcript text: A robot begins at point $C$ and travels in a straight line across a warehouse floor to point A where it picks up some merchandise. It then turns a $37.0^{\circ}$ corner and travels 46.8 m to point B where it drops off the merchandise, as in the figure to the right. If the robot must now tum a $28.0^{\circ}$ corner to return to its original position, how far must it travel to get there? Describe how to solve this problem. To determine the desired distance, use the given information to determine $\square$ and then use the formula $\square$ The robot must travel $\square$ $\square$ to travel to its original position. (Round to one decimal place as needed.)

Solution

Solution Steps

Step 1: Understand the Problem

The robot travels from point C to point A, then turns 37.0° and travels 46.8 m to point B. We need to determine the distance the robot must travel to return to its original position at point C.

Step 2: Identify the Triangle

The robot's path forms a triangle with points A, B, and C. We need to find the length of side BC to determine the distance the robot must travel to return to point C.

Step 3: Use the Law of Cosines

To find the length of side BC, we use the Law of Cosines: \[ BC^2 = AB^2 + AC^2 - 2 \cdot AB \cdot AC \cdot \cos(\angle BAC) \] Given:

$ AB = 46.8 $ m
$ AC = 28.0 $ m
$ \angle BAC = 37.0° $

Step 4: Calculate BC

Substitute the given values into the Law of Cosines formula: \[ BC^2 = 46.8^2 + 28.0^2 - 2 \cdot 46.8 \cdot 28.0 \cdot \cos(37.0°) \] \[ BC^2 = 2190.24 + 784 - 2 \cdot 46.8 \cdot 28.0 \cdot 0.7986 \] \[ BC^2 = 2190.24 + 784 - 2096.35 \] \[ BC^2 = 877.89 \] \[ BC = \sqrt{877.89} \] \[ BC \approx 29.6 \]