Answer :
To solve this problem involving a pyramid with a slant height of 25 meters and a total surface area of 2400 square meters, we need to go step by step through the calculations.
### Part (a): Formula to Calculate the Radius/Side of the Base
For this problem, we will consider the pyramid to have a circular base, which means it behaves like a cone. The formula used to calculate the radius [tex]\( r \)[/tex] of the base of a cone involves the total surface area.
The formula for the total surface area [tex]\( S \)[/tex] of a cone with radius [tex]\( r \)[/tex] and slant height [tex]\( l \)[/tex] is:
[tex]\[ S = \pi r (r + l) \][/tex]
Where:
- [tex]\( S \)[/tex] is the total surface area (2400 m²).
- [tex]\( \pi \)[/tex] is the mathematical constant (approximated as 3.14159).
- [tex]\( r \)[/tex] is the radius of the base.
- [tex]\( l \)[/tex] is the slant height (25 m).
### Part (b): Circumference of the Circular Base
Once we have the radius [tex]\( r \)[/tex], we can find the circumference [tex]\( C \)[/tex] of the circular base using the formula:
[tex]\[ C = 2 \pi r \][/tex]
Given the radius [tex]\( r \)[/tex] is approximately 17.8347 meters, the circumference is calculated as:
[tex]\[ C = 2 \pi \times 17.8347 \approx 112.06 \text{ meters} \][/tex]
### Part (c): Amount to be Filled with Water at Rs. 0.75 per Liter
To find the amount of water needed (in liters) to fill the volume of the pyramid-cone and the total cost, we need to calculate the volume [tex]\( V \)[/tex] of the cone.
The formula for the volume [tex]\( V \)[/tex] of a cone is:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Where [tex]\( h \)[/tex] is the height of the cone.
To find the height [tex]\( h \)[/tex], we use the Pythagorean theorem in the triangle formed by the slant height [tex]\( l \)[/tex], radius [tex]\( r \)[/tex], and height [tex]\( h \)[/tex]:
[tex]\[ l^2 = r^2 + h^2 \][/tex]
[tex]\[ h = \sqrt{l^2 - r^2} \][/tex]
Given:
- Slant height [tex]\( l = 25 \)[/tex] meters
- Radius [tex]\( r \approx 17.8347 \)[/tex] meters
The height [tex]\( h \)[/tex] is calculated as:
[tex]\[ h \approx \sqrt{25^2 - 17.8347^2} \approx 17.8258 \text{ meters} \][/tex]
Using the found values, the volume [tex]\( V \)[/tex] of the cone is:
[tex]\[ V = \frac{1}{3} \pi \times (17.8347)^2 \times 17.8258 \approx 5835.46 \text{ cubic meters} \][/tex]
To convert the volume from cubic meters to liters (1 cubic meter = 1000 liters):
[tex]\[ V \approx 5835461.04 \text{ liters} \][/tex]
Finally, to calculate the cost of filling the pyramid with water at Rs. 0.75 per liter:
[tex]\[ \text{Total Cost} = 5835461.04 \text{ liters} \times 0.75 \text{ Rs/liter} \approx 4376595.78 \text{ Rs} \][/tex]
### Summary:
- Radius of the base: [tex]\( \approx 17.83 \text{ meters} \)[/tex]
- Circumference of the base: [tex]\( \approx 112.06 \text{ meters} \)[/tex]
- Volume of pyramid: [tex]\( \approx 5835.46 \text{ cubic meters} \)[/tex]
- Volume in liters: [tex]\( \approx 5835461.04 \text{ liters} \)[/tex]
- Total cost to fill with water: [tex]\( \approx 4376595.78 \text{ Rs} \)[/tex]
These are the detailed steps and corresponding results for the given problem.
### Part (a): Formula to Calculate the Radius/Side of the Base
For this problem, we will consider the pyramid to have a circular base, which means it behaves like a cone. The formula used to calculate the radius [tex]\( r \)[/tex] of the base of a cone involves the total surface area.
The formula for the total surface area [tex]\( S \)[/tex] of a cone with radius [tex]\( r \)[/tex] and slant height [tex]\( l \)[/tex] is:
[tex]\[ S = \pi r (r + l) \][/tex]
Where:
- [tex]\( S \)[/tex] is the total surface area (2400 m²).
- [tex]\( \pi \)[/tex] is the mathematical constant (approximated as 3.14159).
- [tex]\( r \)[/tex] is the radius of the base.
- [tex]\( l \)[/tex] is the slant height (25 m).
### Part (b): Circumference of the Circular Base
Once we have the radius [tex]\( r \)[/tex], we can find the circumference [tex]\( C \)[/tex] of the circular base using the formula:
[tex]\[ C = 2 \pi r \][/tex]
Given the radius [tex]\( r \)[/tex] is approximately 17.8347 meters, the circumference is calculated as:
[tex]\[ C = 2 \pi \times 17.8347 \approx 112.06 \text{ meters} \][/tex]
### Part (c): Amount to be Filled with Water at Rs. 0.75 per Liter
To find the amount of water needed (in liters) to fill the volume of the pyramid-cone and the total cost, we need to calculate the volume [tex]\( V \)[/tex] of the cone.
The formula for the volume [tex]\( V \)[/tex] of a cone is:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Where [tex]\( h \)[/tex] is the height of the cone.
To find the height [tex]\( h \)[/tex], we use the Pythagorean theorem in the triangle formed by the slant height [tex]\( l \)[/tex], radius [tex]\( r \)[/tex], and height [tex]\( h \)[/tex]:
[tex]\[ l^2 = r^2 + h^2 \][/tex]
[tex]\[ h = \sqrt{l^2 - r^2} \][/tex]
Given:
- Slant height [tex]\( l = 25 \)[/tex] meters
- Radius [tex]\( r \approx 17.8347 \)[/tex] meters
The height [tex]\( h \)[/tex] is calculated as:
[tex]\[ h \approx \sqrt{25^2 - 17.8347^2} \approx 17.8258 \text{ meters} \][/tex]
Using the found values, the volume [tex]\( V \)[/tex] of the cone is:
[tex]\[ V = \frac{1}{3} \pi \times (17.8347)^2 \times 17.8258 \approx 5835.46 \text{ cubic meters} \][/tex]
To convert the volume from cubic meters to liters (1 cubic meter = 1000 liters):
[tex]\[ V \approx 5835461.04 \text{ liters} \][/tex]
Finally, to calculate the cost of filling the pyramid with water at Rs. 0.75 per liter:
[tex]\[ \text{Total Cost} = 5835461.04 \text{ liters} \times 0.75 \text{ Rs/liter} \approx 4376595.78 \text{ Rs} \][/tex]
### Summary:
- Radius of the base: [tex]\( \approx 17.83 \text{ meters} \)[/tex]
- Circumference of the base: [tex]\( \approx 112.06 \text{ meters} \)[/tex]
- Volume of pyramid: [tex]\( \approx 5835.46 \text{ cubic meters} \)[/tex]
- Volume in liters: [tex]\( \approx 5835461.04 \text{ liters} \)[/tex]
- Total cost to fill with water: [tex]\( \approx 4376595.78 \text{ Rs} \)[/tex]
These are the detailed steps and corresponding results for the given problem.
