A circular pond is 26yd in diameter is surrounded by a gravel path 2yd wide. The path is to be replaced by a brick wall costing 50$ per square yard. How much will the walk cost.
How do i show my work
pond area = 169π yd^2
pond+path area = 196π yd^2
so, path area = 27π yd^2
Now get the cost = area * $50/yd^2
assuming the walk costs the same as the "wall"
To calculate the cost of the brick wall for the entire path, we need to find the total area of the path and then multiply it by the cost per square yard.
Let's break down the calculation step-by-step:
1. Calculate the area of the circular pond:
- The pond has a diameter of 26 yards, so the radius is half of that, which is 13 yards.
- The formula for the area of a circle is A = πr², where A represents the area and r represents the radius.
- Plugging in the values, we get A = π(13)².
2. Calculate the area of the circular path:
- The path surrounds the pond on the outside, and it has a uniform width of 2 yards.
- To find the area of the path, we need to subtract the area of the pond from the area of the larger circle formed by the outer edge of the path.
- The radius of the larger circle is the sum of the radius of the pond and the width of the path, which is 13 + 2 = 15 yards.
- So, the area of the path is A = π(15)² - π(13)².
3. Calculate the cost of the brick wall:
- Multiply the area of the path by the cost per square yard, which is $50.
- Cost = Area of the path × Cost per square yard.
Now, let's perform the calculations:
1. Area of the pond:
- A = π(13)² = 169π square yards.
2. Area of the path:
- A = π(15)² - π(13)²
= 225π - 169π
= 56π square yards.
3. Cost of the brick wall:
- Cost = Area of the path × Cost per square yard
= 56π × 50
≈ $2800π (exact value)
≈ $8796.27 (approximate value, rounded to the nearest cent).
Therefore, the cost of the brick wall for the path would be approximately $8796.27.