A goat is tethered to an outside corner of a shed by a rope 10 feet long. The goat cannot enter the shed, which is a rectangular 5 ft by 20 ft, but otherwise is unobstructed and can roam freely across the flat ground.

What is the area of the land in which the goat can roam?
answer in ft²

To calculate the area in which the goat can roam, we need to consider the different parts of the goat's movement around the shed.

1. **Goat's movement from the corner:**
   The goat is tethered to an outside corner of a rectangular shed with dimensions 5 ft by 20 ft, and the length of the rope is 10 ft. Since the goat is tethered at an outside corner, its movement will trace part of a circle, but some of the area it could normally roam is blocked by the shed.

### Step 1: Calculate the first quarter-circle area
When tethered to a corner of the shed, the goat can roam freely in a quarter-circle with radius 10 feet from that corner. The area of a quarter-circle is:

\[
A_1 = \frac{1}{4} \pi r^2 = \frac{1}{4} \pi (10)^2 = 25 \pi \, \text{ft}^2
\]

### Step 2: Calculate the additional area along the two sides
The rope can reach beyond the edge of the shed on both the shorter (5 ft) and longer (20 ft) sides of the shed.

- Along the 5 ft side, the rope can reach an additional \(10 - 5 = 5\) feet beyond the corner.
  The goat can roam a quarter-circle with radius 5 feet from this point. The area of this quarter-circle is:

\[
A_2 = \frac{1}{4} \pi (5)^2 = \frac{25 \pi}{4} = 6.25 \pi \, \text{ft}^2
\]

- Along the 20 ft side, the rope can reach an additional \(10 - 20 = -10\) feet, but since 10 ft is less than the length of the shed, the goat cannot reach this side.

### Step 3: Total roaming area
The total area the goat can roam is the sum of the areas of the quarter-circles calculated above:

\[
A_{\text{total}} = A_1 + A_2 = 25 \pi + 6.25 \pi = 31.25 \pi \, \text{ft}^2
\]

Approximating \( \pi \approx 3.1416 \), we get:

\[
A_{\text{total}} \approx 31.25 \times 3.1416 \approx 98.17 \, \text{ft}^2
\]

Thus, the area in which the goat can roam is approximately **98.17 ft²**.

Would you like further details or clarifications?

Here are 5 follow-up questions:
1. How would the area change if the rope were 15 feet long instead of 10?
2. What if the shed were a different shape, such as circular—how would that impact the calculation?
3. Can you calculate the area if the goat were tethered at the middle of one side of the shed instead of a corner?
4. How do you calculate the area if there are obstacles other than the shed?
5. What would be the roaming area if the goat were tethered to the inside of the shed?

**Tip:** Understanding the geometry of shapes, such as quarter-circles, is key when dealing with partial movements around obstructions like buildings.

A goat is tethered to an outside corner of a shed by a rope 10 feet long. The goat cannot enter the shed, which is a rectangular 5 ft by 20 ft, but otherwise is unobstructed and can roam freely across the flat ground. What is the area of the land in which the goat can roam? Answer in ft²

Calculate the area in which a goat tethered to the corner of a 5x20 ft shed can roam, using geometry and circular area formulas. Learn step-by-step how to find the total roaming area, approximately 98.17 ft². Ideal for Grades 9-12 students learning geometry.

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