A right circular cylindrical tank of radius *r* cm and height *(r + 12)* cm contains milk. The entire quantity of milk is taken out from the cylindrical tank and poured into *N* number of hemispherical bowls such that each bowl is filled up to its maximum capacity. If the maximum capacity of each bowl is $$\frac{11r^3}{35} ext{ cm}^3$$, which among the following (I, II, III & IV) is/are the possible value(s) of *N*? (*r* and *N* are positive integers)

I. 6
II. 34
III. 25
IV. 19

Question

A right circular cylindrical tank of radius *r* cm and height *(r + 12)* cm contains milk. The entire quantity of milk is taken out from the cylindrical tank and poured into *N* number of hemispherical bowls such that each bowl is filled up to its maximum capacity. If the maximum capacity of each bowl is $$\frac{11r^3}{35} 	ext{ cm}^3$$, which among the following (I, II, III & IV) is/are the possible value(s) of *N*? (*r* and *N* are positive integers)

I. 6  
II. 34  
III. 25  
IV. 19

Accepted Answer

**Solution:**

1. Volume of the cylindrical tank:
   $$V_{cyl} = \pi r^2 h = \frac{22}{7} 	imes r^2 	imes (r + 12)$$

2. Volume of each hemispherical bowl:
   $$V_{bowl} = \frac{11 r^3}{35}$$

3. Total volume of milk in the tank is poured into *N* bowls:
   $$V_{cyl} = N 	imes V_{bowl}$$

4. Substitute values:
   $$\frac{22}{7} r^2 (r + 12) = N 	imes \frac{11 r^3}{35}$$

5. Simplify the equation:
   Multiply both sides by 35 to clear denominator:
   $$35 	imes \frac{22}{7} r^2 (r + 12) = 11 r^3 N$$

$$5 	imes 22 r^2 (r + 12) = 11 r^3 N$$

$$110 r^2 (r + 12) = 11 r^3 N$$

6. Divide both sides by 11 r^2 (r > 0):
   $$10 (r + 12) = r N$$

7. Rearranged:
   $$10 r + 120 = r N$$

8. Divide both sides by r:
   $$10 + \frac{120}{r} = N$$

9. Now check each given *N* value for positive integer *r*:

- For I. *N = 6*:
     $$6 = 10 + \frac{120}{r} \Rightarrow \frac{120}{r} = -4 \Rightarrow r = -30$$ (Not possible as *r* > 0)

- For II. *N = 34*:
     $$34 = 10 + \frac{120}{r} \Rightarrow \frac{120}{r} = 24 \Rightarrow r = 5$$ (Valid)

- For III. *N = 25*:
     $$25 = 10 + \frac{120}{r} \Rightarrow \frac{120}{r} = 15 \Rightarrow r = 8$$ (Valid)

- For IV. *N = 19*:
     $$19 = 10 + \frac{120}{r} \Rightarrow \frac{120}{r} = 9 \Rightarrow r = \frac{120}{9} = \frac{40}{3}$$ (Not an integer)

**Answer:** Both II & III are possible values of *N*.

Answer

Answer

Answer

Answer