Let's go through the solution step-by-step:
### (a) Write the formula to find the volume of the cone:
Given a cone, the volume [tex]\( V_{\text{cone}} \)[/tex] can be found using the formula:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the base,
- [tex]\( h \)[/tex] is the height of the cone.
### (b) How much water does the tank hold?
To determine how much water the tank can hold, let's break it down into the volumes of the cylindrical part and the conical part.
1. Volume of the conical part:
Given:
- Base area ([tex]\( A \)[/tex]) = 38.5 m²
- Height of the cone ([tex]\( h_{\text{cone}} \)[/tex]) = 6 m
First, convert the base area to square centimeters for consistency:
[tex]\[ A = 38.5 \, \text{m}^2 = 38.5 \times 10,000 \, \text{cm}^2 = 385,000 \, \text{cm}^2 \][/tex]
Now, the height in centimeters:
[tex]\[ h_{\text{cone}} = 6 \, \text{m} = 600 \, \text{cm} \][/tex]
Using the volume formula for the cone and the given values, we have:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \times 385,000 \, \text{cm}^2 \times 600 \, \text{cm} = 77,000,000 \, \text{cm}^3 \][/tex]
2. Volume of the cylindrical part:
Given:
- Total height of the tank ([tex]\( h_{\text{total}} \)[/tex]) = 14 m
- Height of the cylindrical part ([tex]\( h_{\text{cylinder}} \)[/tex]) = Total height - Height of the cone
- Base area remains the same.
Calculate the height in meters:
[tex]\[ h_{\text{cylinder}} = 14 \, \text{m} - 6 \, \text{m} = 8 \, \text{m} \][/tex]
Convert it to centimeters:
[tex]\[ h_{\text{cylinder}} = 8 \, \text{m} = 800 \, \text{cm} \][/tex]
Using the volume formula for the cylinder:
[tex]\[ V_{\text{cylinder}} = 385,000 \, \text{cm}^2 \times 800 \, \text{cm} = 308,000,000 \, \text{cm}^3 \][/tex]
3. Total volume of the tank:
Sum up the volumes:
[tex]\[ V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}} = 77,000,000 \, \text{cm}^3 + 308,000,000 \, \text{cm}^3 = 385,000,000 \, \text{cm}^3 \][/tex]
4. Convert the total volume to liters (since [tex]\( 1 \, \text{cm}^3 = 0.001 \, \text{liters} \)[/tex]):
[tex]\[ V_{\text{total}} = 385,000,000 \, \text{cm}^3 \times 0.001 \, \text{liters/cm}^3 = 385,000 \, \text{liters} \][/tex]
### (c) If the tank is filled with water at the rate of 24 paisa per litre, what is the total cost?
Given:
- Cost per liter = 24 paisa = 0.24 Rs
Calculate total cost:
[tex]\[ \text{Total Cost} = \text{Volume in liters} \times \text{Cost per liter} = 385,000 \, \text{liters} \times 0.24 \, \text{Rs} = 92,400 \, \text{Rs} \][/tex]
### Summary:
(a) Formula to find the volume of the cone:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \][/tex]
(b) The tank holds:
[tex]\[ 385,000 \, \text{liters} \][/tex]
(c) The total cost to fill the tank:
[tex]\[ Rs. 92,400 \][/tex]
