In a marriage ceremony of her daughter Poonam, Ashok has to make arrangements for the accommodation of 150 persons. For this purpose, he plans to build a conical tent in such a way that each person have 4 sq. metres of the space on ground and 20 cubic metres of air to breath. What should be the height of the conical tent?

Options

A. 20 cm
B. 15 cm
C. 12 cm
D. 30

Answer

To find the height of the conical tent that Ashok needs to build for accommodating 150 persons, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the total ground area required for 150 persons:**
Each person requires 4 square meters of space. Therefore, for 150 persons:
\[ \text{Total area} = 150 \times 4 = 600 \text{ square meters} \]

**Hint:** Multiply the number of persons by the area required per person to find the total area.

2. **Set up the formula for the area of the base of the cone:**
The area of the base of a cone is given by the formula:
\[ \text{Area} = \pi r^2 \]
We know the total area is 600 square meters, so:
\[ \pi r^2 = 600 \]

**Hint:** Use the area formula for a circle to relate it to the total area required.

3. **Substitute the value of \(\pi\) and solve for \(r^2\):**
Using \(\pi \approx \frac{22}{7}\):
\[ \frac{22}{7} r^2 = 600 \]
Multiply both sides by 7:
\[ 22 r^2 = 4200 \]
Now divide by 22:
\[ r^2 = \frac{4200}{22} = 190.91 \text{ (approximately)} \]

**Hint:** Rearranging the equation helps isolate \(r^2\).

4. **Calculate the total volume of air required for 150 persons:**
Each person requires 20 cubic meters of air. Therefore, for 150 persons:
\[ \text{Total volume} = 150 \times 20 = 3000 \text{ cubic meters} \]

**Hint:** Multiply the number of persons by the volume required per person to find the total volume.

5. **Set up the formula for the volume of the cone:**
The volume \(V\) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
We know the volume is 3000 cubic meters, so:
\[ \frac{1}{3} \pi r^2 h = 3000 \]

**Hint:** Use the volume formula for a cone to relate it to the total volume required.

6. **Substitute the value of \(\pi\) and \(r^2\) into the volume formula:**
Substituting \(\pi\) and \(r^2\):
\[ \frac{1}{3} \times \frac{22}{7} \times 190.91 \times h = 3000 \]
Rearranging gives:
\[ h = \frac{3000 \times 3 \times 7}{22 \times 190.91} \]

**Hint:** Isolate \(h\) by multiplying both sides by the reciprocal of the other terms.

7. **Calculate the height \(h\):**
Performing the calculations:
\[ h = \frac{63000}{4199.99} \approx 15 \text{ meters} \]

**Hint:** Ensure to perform the arithmetic carefully to find the height.

### Final Answer:
The height of the conical tent should be **15 meters**.