Dakota the dog is tied to a 6m leash on a corner of a 4m by 5m shed.  How much area does Dakota have in which to play if he can only go around the outside of the shed? 

To determine the area Dakota can play in, we need to account for the fact that the dog is tied to a 6-meter leash at a corner of the shed and can only move around the outside of the shed.

### Step 1: Analyze the Movement Around the Shed

- The shed has dimensions of 4 meters by 5 meters.
- Since Dakota can only go around the outside of the shed, we’ll break his movement into two parts:
  1. When the dog is on the longer side (5 meters).
  2. When the dog is on the shorter side (4 meters).

### Step 2: Calculate the Playable Area
Dakota has 6 meters of leash, which allows him to trace part of a circle around the shed's corner. He can move around two sides of the shed, creating different segments of circles at each corner. Let’s break it down:

#### 1. **Full Semi-Circle Play Area**
At the corner of the shed, Dakota can trace out a **quarter-circle** with a radius of 6 meters. The area of a quarter-circle is:

\[
\text{Area of quarter-circle} = \frac{1}{4} \times \pi \times r^2 = \frac{1}{4} \times \pi \times (6)^2 = \frac{1}{4} \times \pi \times 36 = 9\pi \, \text{square meters}
\]

#### 2. **Other Usable Areas**
When Dakota moves along the 4-meter and 5-meter sides, the leash is extended away from the shed, giving him access to a **quarter-circle** around each corner. These partial areas overlap with his movement from the main quarter-circle.

### Total Area
The total area available for Dakota to play in is the sum of the area along both sides and around the corners.

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Dakota the dog is tied to a 6m leash on a corner of a 4m by 5m shed. How much area does Dakota have in which to play if he can only go around the outside of the shed?

Determine how much area Dakota the dog has to play when tied to a 6m leash at the corner of a 4m by 5m shed. This problem involves calculating areas of quarter-circles and understanding geometric principles. Suitable for Grades 7-10.

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