A dog is attached to a 21-foot rope fastened to the outside corner of a fenced-in garden that measures 18 feet by 22 feet. Assuming that the dog cannot enter the garden, compute the exact area that the dog can wander.
I'm confused as to how they want me to solve this. I couldn't get the diagram to look right.
the rope reaches around the corner on the 18 ft side, but not on the 22 ft side
so the area is half of a 42 ft dia circle, plus a quarter of a 6 ft dia circle
To compute the exact area that the dog can wander, we can visualize the situation as follows:
1. The dog is attached to a 21-foot rope.
2. The rope is fastened to the outside corner of the fenced-in garden.
3. The garden measures 18 feet by 22 feet.
We need to determine the area that the dog can reach while being confined by the rope and unable to enter the garden.
To solve this, we can follow these steps:
Step 1: Draw a diagram
- Draw a rectangle to represent the fenced-in garden, with dimensions 18 feet by 22 feet.
- Label the outside corner where the rope is attached as point A.
Step 2: Determine the effective radius of the dog's wandering area
- Since the dog is confined by the rope, its wandering area will form a circular region.
- The radius of this circle is equal to the length of the rope, which is 21 feet.
Step 3: Draw the circular wandering area
- From point A, draw a circle with a radius of 21 feet. This represents the area the dog can wander.
Step 4: Calculate the area of the circular wandering area
- Use the formula for the area of a circle: A = πr^2, where A is the area and r is the radius.
- Substituting the radius with 21 feet, the area is A = π(21^2).
Step 5: Simplify and calculate the area
- Simplify the expression by squaring the radius: A = π(441).
- Multiply the constant π with 441: A = 441π.
- The area of the dog's wandering area is equal to 441π square feet.
So, the exact area that the dog can wander is 441π square feet or approximately 1383.48 square feet.
To solve this problem, we need to determine the shape of the area that the dog can wander within the garden. Since the dog is attached to a 21-foot rope, it can move in any direction within a 21-foot radius from the point where it is attached to the garden's corner.
First, let's consider the possible positions for the dog's rope attachment point. The garden is a rectangle with dimensions of 18 feet by 22 feet. The attachment point can be located anywhere along the 18-foot side or the 22-foot side of the garden.
To visualize this, you can draw a rectangle to represent the garden and mark a point at one of the corners as the attachment point of the dog's rope. Then, draw a circle with a radius of 21 feet centered at the attachment point. The dog can wander inside this circle.
Next, we need to determine the exact area within the circle that the dog can wander. The shape of this area is a sector of a circle, bounded by the circle's circumference and two radii.
To calculate the area of the sector, we need the measure of the sector's central angle. This angle is formed by the two radii that connect the attachment point to the endpoints of the circle's circumference where the dog can wander.
To find the central angle, we can use trigonometry. The length of the adjacent side is half of the length of the garden's side along which the attachment point lies (9 feet or 11 feet), and the hypotenuse is the radius of the circle (21 feet). Using arccosine, we can find the angle.
Once we have the central angle, we can calculate the area of the sector using the formula: Area of sector = (central angle / 360 degrees) * pi * (radius)^2.
Remember to multiply the resulting area by 4, as there are four attachment points in total (one at each corner of the garden).
By following these steps, you should be able to compute the exact area that the dog can wander within the garden.