https://file.aiquickdraw.com/mathbot/file/172944482669264zuvhsq.jpg

Let's work through the problems from the image step by step:

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

#### Given:
- Radius (r) = 4' (this is from the diagram in the question)
- Height (h) = not directly given (you might need to clarify this detail based on additional text or measurement).

The formula for the volume of a cylinder is:
\[
V = \pi r^2 h
\]
Without knowing the height, I can't calculate the volume for this specific cistern yet.

Could you provide the missing height for the cistern or more details from the problem? Or I can assist with another part of the exercise.


### Problem 2:
How many cubic yards of earth are excavated for a concrete cistern that has:
- Diameter = 9'-6"
- Depth = 10'-0" (outside dimensions)
- An allowance of 18 inches made around the walls for form work.
- The top of the cistern is 4'-0" below the grade.

#### Step-by-step:
1. Convert dimensions to feet:
   - Diameter of the cistern: 9'-6" = 9.5 feet
   - Radius (r) = 9.5 / 2 = 4.75 feet
   - Depth (h) = 10 feet
   
   Allowance of 18 inches = 1.5 feet.
   
2. Add allowance to the diameter:
   - New diameter = 9.5 + 2(1.5) = 9.5 + 3 = 12.5 feet
   - New radius (with allowance) = 12.5 / 2 = 6.25 feet
   
3. The volume of the cylindrical excavation (with the allowance included) is:
   \[
   V = \pi r^2 h = 3.1416 \times (6.25^2) \times 10
   \]
   First, calculate \(6.25^2\):
   \[
   6.25^2 = 39.0625
   \]
   Now calculate the volume:
   \[
   V = 3.1416 \times 39.0625 \times 10 = 1226.56 \text{ cubic feet}
   \]
   Convert this to cubic yards (since there are 27 cubic feet in a cubic yard):
   \[
   V = \frac{1226.56}{27} \approx 45.42 \text{ cubic yards}
   \]
   So, approximately **45.42 cubic yards** of earth are excavated for the cistern.

---

Would you like more details on these calculations or further clarification? Here are 5 related questions you might explore:

1. How would the volume change if the depth of the cistern increased by 2 feet?
2. What would be the volume in cubic meters instead of cubic yards?
3. How does increasing the radius of a cylinder affect its volume?
4. What is the difference between inside and outside dimensions in volume calculations?
5. How would an increase in the allowance for form work impact the total volume of earth excavated?

**Tip**: Always ensure unit conversions (feet, inches, yards) are accurate before performing volume calculations to avoid 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"?

Solve volume measurement problems for cylindrical cisterns using key geometry concepts. Learn how to calculate the volume of a cistern and determine how much earth is excavated for a cylindrical shape. Suitable for Grades 8-10 students.

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