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Write the question out and solve the answer

### Questions and Solutions

#### Review Problem 1:
1. **Problem:** A cross-section of a cylindrical cistern is shown. What is the volume of the cistern?

   **Solution:**
   To calculate the volume of a cylinder, we use the formula:
   \[
   V = \pi r^2 h
   \]
   where:
   - \( \pi = 3.1416 \),
   - \( r \) is the radius of the circular end, and
   - \( h \) is the height (or depth) of the cylinder.

   From the image:
   - The diameter of the cistern is \( 2' \), so the radius \( r = \frac{2}{2} = 1' \).
   - The depth (height) of the cistern \( h = 4' \).

   Plugging in these values:
   \[
   V = 3.1416 \times (1)^2 \times 4
   \]
   \[
   V = 3.1416 \times 1 \times 4
   \]
   \[
   V = 12.5664 \text{ cubic feet}
   \]
   The volume of the cistern is **12.57 cubic feet** (rounded to the nearest hundredth).

---

#### Review Problem 2:
2. **Problem:** How many cubic yards of earth are excavated for a concrete cistern that has a diameter of \(9'6"\) and a depth of \(10'0"\) as outside dimensions? An allowance of 18 inches is made around the walls for form work. The top of the cistern is \(4'0"\) below the grade.

   **Solution:**
   To solve this, we need to calculate the volume of the excavation area, which includes the allowance for formwork.

   - Convert all dimensions to feet:
     - Diameter of cistern = \(9'6" = 9.5\) feet, so radius \(r = \frac{9.5}{2} = 4.75\) feet.
     - Depth (height) of the cistern \(h = 10\) feet.
     - Allowance around walls for formwork = 18 inches = \(1.5\) feet.

   - Add the formwork allowance to both the diameter and height of the cistern:
     - New diameter with formwork = \(9.5 + 1.5 + 1.5 = 12.5\) feet, so new radius \(= \frac{12.5}{2} = 6.25\) feet.
     - New height with formwork \(= 10 + 1.5 = 11.5\) feet.

   - Calculate the volume of the excavation:
     \[
     V = \pi r^2 h
     \]
     \[
     V = 3.1416 \times (6.25)^2 \times 11.5
     \]
     \[
     V = 3.1416 \times 39.0625 \times 11.5
     \]
     \[
     V = 1409.5767 \text{ cubic feet}
     \]

   - Convert cubic feet to cubic yards (since 1 cubic yard = 27 cubic feet):
     \[
     \text{Volume in cubic yards} = \frac{1409.5767}{27}
     \]
     \[
     \approx 52.21 \text{ cubic yards}
     \]
   Approximately **52.21 cubic yards** of earth are excavated.

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Would you like further details on any steps or clarification on specific parts of the solutions?

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### Follow-up Questions:
1. How would the volume change if the depth of the cistern was increased by 2 feet?
2. How would you calculate the surface area of the cylindrical cistern?
3. If the diameter of the cistern in Problem 1 were doubled, what would be the new volume?
4. How does changing the value of \( \pi \) to 3.14 affect the calculations?
5. How can the formula for the volume of a cylinder be derived from basic principles?

**Tip:** Always ensure units are consistent before solving volume or area problems to avoid conversion errors.

1. A cross-section of a cylindrical cistern is shown. What is the volume of the cistern?
2. How many cubic yards of earth are excavated for a concrete cistern that has a diameter of 9'6" and a depth of 10'0" as outside dimensions? An allowance of 18 inches is made around the walls for form work. The top of the cistern is 4'0" below the grade.

Solve volume and excavation problems for a cylindrical cistern with a radius and height provided. Learn to calculate the volume of a cistern using V = πr^2h and convert cubic feet to cubic yards. Suitable for Grades 10-12.

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