Given the following:

- (i) X is a cone with volume 1232 m³
- (ii) Y is a cylinder with volume 1848 m³
- (iii) S is a square inscribed in a circle

Column II:
- (A) Radius = 14 m
- (B) Radius = 7 m
- (C) Circumference of circle = 44 m

If the difference between the height of Y and the side of S is greater than 20 m, then which of the following options correctly matches the columns for the given conditions?

(a) (i) Only A, either B or C (ii) Only A (iii) Only A
(b) (i) Only C (ii) Only B (iii) Only A
(c) (i) Only B (ii) Only A (iii) Only A
(d) (i) Only C (ii) Only B (iii) Only B
(e) None of these

Question

Given the following:

- (i) X is a cone with volume 1232 m³
- (ii) Y is a cylinder with volume 1848 m³
- (iii) S is a square inscribed in a circle

Column II:
- (A) Radius = 14 m
- (B) Radius = 7 m
- (C) Circumference of circle = 44 m

If the difference between the height of Y and the side of S is greater than 20 m, then which of the following options correctly matches the columns for the given conditions?

(a) (i) Only A, either B or C (ii) Only A (iii) Only A
(b) (i) Only C (ii) Only B (iii) Only A
(c) (i) Only B (ii) Only A (iii) Only A
(d) (i) Only C (ii) Only B (iii) Only B
(e) None of these

Accepted Answer

**Solution:**

1. From the given circumference of the circle (C) in Column II (C = 44 m), calculate the radius:
   $$2 	imes \frac{22}{7} 	imes r = 44 \implies r = 7 	ext{ m}$$

2. Since the square S is inscribed in the circle, the side of the square (s) is twice the radius:
   $$s = 2 	imes 7 = 14 	ext{ m}$$

3. For the cylinder Y with volume 1848 m³:
   Volume of cylinder = $$\pi r^2 h_y$$
   Using radius 7 m:
   $$\frac{22}{7} 	imes 7^2 	imes h_y = 1848 \implies 22 	imes 7 	imes h_y = 1848 \implies h_y = \frac{1848}{154} = 12 	ext{ m}$$

4. Calculate the difference between height of Y and side of S:
   $$|h_y - s| = |12 - 14| = 2 	ext{ m}$$
   This is not greater than 20 m, so radius 7 m does not satisfy the condition.

5. Now consider radius 14 m:
   - Side of square:
     $$s = 2 	imes 14 = 28 	ext{ m}$$
   - Height of cylinder Y:
     $$\frac{22}{7} 	imes 14^2 	imes h_y = 1848 \implies 22 	imes 14 	imes h_y = 1848 \implies h_y = \frac{1848}{308} = 6 	ext{ m}$$

6. Difference:
   $$|h_y - s| = |6 - 28| = 22 	ext{ m} > 20$$
   This satisfies the condition.

7. For the cone X with volume 1232 m³ and radius 14 m:
   Volume of cone = $$\frac{1}{3} \pi r^2 h_x$$
   $$1232 = \frac{1}{3} 	imes \frac{22}{7} 	imes 14^2 	imes h_x$$
   $$1232 = \frac{1}{3} 	imes \frac{22}{7} 	imes 196 	imes h_x$$
   $$1232 = \frac{1}{3} 	imes 616 	imes h_x$$
   $$1232 = 205.33 	imes h_x$$
   $$h_x = \frac{1232}{205.33} \approx 6 	ext{ m}$$

8. Based on these calculations:
   - (i) X = Cone with radius 14 m (A)
   - (ii) Y = Cylinder with radius 7 m (B) or 14 m (A) depending on volume
   - (iii) S = Square inscribed in circle with circumference 44 m (C)

9. From the problem's conditions and calculations, the correct matching is:
   - (i) Only A
   - (ii) Only A
   - (iii) Only A

**Answer:** (a) (i) Only A, either B or C (ii) Only A (iii) Only A

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