Let's tackle each question in the "Review Problems" section step-by-step.

Problem 1:

We are given:

• Diameter of the cistern = 4'0" (or 4 feet)
• Therefore, the radius $r = \frac{4}{2} = 2$ feet.

Using the volume formula for a cylinder: $V = \pi r^2 h$ Substitute the values:

• Radius $r = 2$ feet
• Height $h = 8$ feet

Using $\pi = 3.1416$: $V = 3.1416 \times 2^2 \times 8$ $V = 3.1416 \times 4 \times 8$ $V = 3.1416 \times 32$ $V \approx 100.53 \text{ cubic feet}$

So, the volume of the cistern is approximately 100.53 cubic feet.

Problem 2:

For this problem:

• Outside diameter = 9'6" = 9.5 feet
• Outside depth = 10'0" = 10 feet
• Allowance for form work (walls) is 18 inches (1.5 feet), meaning we’ll reduce 1.5 feet from the diameter and depth for the actual inside measurements.

Step 1: Find the effective inner diameter and height.

• Effective inner diameter = 9.5 - 2(1.5) = 6.5 feet
  • Radius $r = \frac{6.5}{2} = 3.25$ feet
• Effective inner depth (height) = 10 - 1.5 = 8.5 feet

Step 2: Calculate the volume.

Using the formula: $V = \pi r^2 h$ where:

• $\pi = 3.1416$
• $r = 3.25$
• $h = 8.5$

$V = 3.1416 \times (3.25)^2 \times 8.5$ $V = 3.1416 \times 10.5625 \times 8.5$ $V \approx 282.74 \text{ cubic feet}$

Step 3: Convert to cubic yards.

Since there are 27 cubic feet in a cubic yard: $\frac{282.74}{27} \approx 10.47 \text{ cubic yards}$

So, the amount of earth to be excavated is approximately 10.47 cubic yards.

Would you like a deeper breakdown of these calculations or have any specific questions?

Related Questions:

1. How would you calculate the volume of a cylinder if the radius and height were given in different units (e.g., inches and feet)?
2. If the cistern in Problem 1 had an irregular cross-section, how would the volume formula change?
3. How does the volume calculation change if you need to account for a slanted top or bottom in a cylinder?
4. What is the impact of using different approximations for $\pi$ on the volume results?
5. How would adding a hemispherical dome on top of the cistern affect the total volume?

Tip: Always ensure units are consistent throughout the calculation to avoid conversion errors.