Each of the following question is followed by two statements I , and II. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question.
What is the volume of conical tent?
I. the height and radius of tent is in ratio 4:3, where sum of radius and height is 14m.
II. the slant height is 13 cm while radius is 5 cm.

Options

A. Only Statement I alone.
B. Only Statement II alone.
C. Both Statements I and II together.
D. Statement I alone or Stamtement II alone is sufficient to answer this question

Answer

To solve the problem of finding the volume of the conical tent, we will analyze both statements provided and determine their sufficiency.

### Step 1: Understand the formula for the volume of a cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.

### Step 2: Analyze Statement I
**Statement I:** The height and radius of the tent are in the ratio \( 4:3 \), where the sum of the radius and height is \( 14 \) m.

1. Let the radius \( r \) be \( 3x \) and the height \( h \) be \( 4x \).
2. According to the statement, we have:
\[ r + h = 14 \implies 3x + 4x = 14 \implies 7x = 14 \implies x = 2 \]
3. Therefore, the radius \( r = 3x = 6 \) m and the height \( h = 4x = 8 \) m.

### Step 3: Calculate the volume using Statement I
Now we can substitute the values of \( r \) and \( h \) into the volume formula:
\[ V = \frac{1}{3} \pi (6^2)(8) = \frac{1}{3} \pi (36)(8) = \frac{1}{3} \pi (288) = 96\pi \text{ m}^3 \]
Thus, Statement I is sufficient to determine the volume of the conical tent.

### Step 4: Analyze Statement II
**Statement II:** The slant height is \( 13 \) cm while the radius is \( 5 \) cm.

1. We need to find the height \( h \) using the Pythagorean theorem:
- Let the height be \( h \).
- The slant height \( l = 13 \) cm and the radius \( r = 5 \) cm.
- According to the Pythagorean theorem:
\[ l^2 = r^2 + h^2 \implies 13^2 = 5^2 + h^2 \implies 169 = 25 + h^2 \implies h^2 = 144 \implies h = 12 \text{ cm} \]

### Step 5: Calculate the volume using Statement II
Now we can substitute the values of \( r \) and \( h \) into the volume formula:
\[ V = \frac{1}{3} \pi (5^2)(12) = \frac{1}{3} \pi (25)(12) = \frac{1}{3} \pi (300) = 100\pi \text{ cm}^3 \]
Thus, Statement II is also sufficient to determine the volume of the conical tent.

### Conclusion
Both statements are sufficient to answer the question independently. Therefore, the answer is:

**Option D:** Statement I alone or Statement II alone is sufficient to answer the question.

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