(iv) The total surface area of a pyramid with a slant height of 25 m is 2400 m².

(a) Which formula is used to calculate the radius/side of the base of the given solids?

(b) What is the circumference of the circular base of the solid? Find it.

(c) Calculate the amount to be filled with water at Rs. 0.75 per litre.



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.