The worksheet contains two main problems:

1. Shortest Path Problem: You are asked to evaluate the three proposed paths to go from point $A$ to $B$ (with $AB = 8 \, \text{cm}$), calculate their respective lengths in terms of $\pi$, and determine the shortest path.

   • Path 1: Half a circle.
   • Path 2: Two quarter circles.
   • Path 3: Four smaller arcs.

   To solve this:

   • Calculate the circumference of each path.
   • Use the arc length formula $\text{Arc Length} = \text{radius} \times \text{angle in radians}$ for each segment.

2. Goat's Grazing Area Problem: A goat is tied outside a square enclosure (1m x 1m) with a rope of 5 meters. You are tasked to shade the area of the field that the goat can graze, ensuring the grazing area does not overlap the enclosure.

   • Calculate the grazing area using the area of circular segments, subtracting the inaccessible portions near the enclosure.

Would you like me to provide detailed calculations for either or both problems?

5 Related Questions:

1. What is the formula for calculating arc lengths and how can it be applied here?
2. How do circular and semi-circular grazing areas contribute to the solution of the goat problem?
3. What strategies can be used to compare the three paths in the shortest path problem?
4. How can we calculate the area of circular segments for irregular grazing regions?
5. Why is it important to represent arc lengths in terms of $\pi$ instead of decimals?

Tip:

When solving arc or circle problems, always carefully identify the radius and portion of the circle involved (e.g., half, quarter).