To find the total amount of rock fill required for the four wells, we’ll calculate the volume of each well and then convert the total volume from cubic feet to cubic yards.

1. Determine the Volume for Each Well

   The formula for the volume of a cylinder (which we’ll use here for each well) is: $V = \pi r^2 h$ where:

   • $V$ is the volume,
   • $r$ is the radius,
   • $h$ is the height (or depth) of the well.

2. Calculate Volume of Wells with Diameter 4'-0" and Depth 4'-6"

   • Diameter = 4'-0" $\Rightarrow$ Radius $r = \frac{4}{2} = 2$ feet
   • Depth $h = 4'-6" = 4.5$ feet

   Plugging into the formula: $V = \pi (2)^2 (4.5) = \pi \times 4 \times 4.5 = 18\pi \approx 56.55 \text{ cubic feet}$

   Since there are two of these wells, their combined volume is: $2 \times 56.55 = 113.1 \text{ cubic feet}$

3. Calculate Volume of Wells with Diameter 3'-6" and Depth 5'-0"

   • Diameter = 3'-6" $\Rightarrow$ Radius $r = \frac{3.5}{2} = 1.75$ feet
   • Depth $h = 5$ feet

   Plugging into the formula: $V = \pi (1.75)^2 (5) = \pi \times 3.0625 \times 5 = 15.3125\pi \approx 48.12 \text{ cubic feet}$

   Since there are two of these wells, their combined volume is: $2 \times 48.12 = 96.24 \text{ cubic feet}$

4. Calculate the Total Volume in Cubic Feet

   Adding the volumes of all four wells: $113.1 + 96.24 = 209.34 \text{ cubic feet}$

5. Convert Cubic Feet to Cubic Yards

   Since there are 27 cubic feet in a cubic yard: $\text{Total cubic yards} = \frac{209.34}{27} \approx 7.75 \text{ cubic yards}$

Answer:

The contractor will need approximately 7.75 cubic yards of rock fill for the four wells.

Would you like a breakdown of any specific steps or further details?

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Here are some related questions you might explore:

1. How would the calculation change if one well was deeper than the others?
2. What if the diameter was given in inches instead of feet?
3. How can we use unit conversions more efficiently in different volume calculations?
4. What alternative methods are there for finding the volume of irregular shapes?
5. How does this calculation change if the wells are not perfectly cylindrical?

Tip: Always double-check your unit conversions when dealing with volumes, as these can often cause miscalculations in construction projects.