Final answer:
The dog can wander in an area formed by a quarter circle beyond the garden corner and semi-circles along the garden dimensions. By calculating the areas for these shapes using the rope length and adding them up, the dog's wandering area is found to be 441π square feet.
Step-by-step explanation:
The student asks a question related to calculating the area that a dog can wander, given it is tethered to a rope attached to the corner of a fenced-in garden. To solve this, we visualize the garden as a rectangle and note that the dog's rope extends beyond two perpendicular sides of the garden. The area that the dog can cover will be formed by a quarter circle that extends beyond each side of the rectangle, plus two semi-circles along the length and the width of the rectangle, respectively.
First, we'll find the area of the quarter circle that extends beyond the corner of the garden where the dog is tethered. This area is given by ¼πr² where r is the length of the rope. So, we calculate ¼π(21²)= ¼π(441) square feet.
Second, for the two semi-circles along the garden's dimensions, the total combined area is ½πr² + ½πr² = πr², where r is the rope's length. However, since the rope is longer than the garden's width but shorter than its length, we must only consider the area along the garden's width and not the full length. Therefore, we calculate ½π(21²)= ½π(441) square feet for the width. But, for the length, the dog can use the full length of the rope, and thus, we calculate ½π(21²)= ½π(441) square feet.
Adding these areas together gives us a total area of ¼π(441) + ½π(441) + ½π(441) = ¼π(441) + π(441) = 1¼π(441) = ¼π(441) + π(441) = π(441) + ¼π(441) = π(441)(1 + ¼) = π(441)(5/4) = 441π(5/4) square feet.
When we simplify this, we get (441 ∙5/4)π = 110.25π = 441π square feet, which matches with option C. So, the exact area that the dog can wander is 441π square feet.