A cylindrical vessel with radius of $17.5 ext{ cm}$ and height of $18 ext{ cm}$ is filled up to $80\%$ of its capacity with milk. If the total milk from the cylindrical vessel is transferred into $30$ cuboidal vessels whose length and breadth are $7 ext{ cm}$ and $3 ext{ cm}$ respectively, find the height of each cuboidal vessel.

Question

A cylindrical vessel with radius of $17.5 	ext{ cm}$ and height of $18 	ext{ cm}$ is filled up to $80\%$ of its capacity with milk. If the total milk from the cylindrical vessel is transferred into $30$ cuboidal vessels whose length and breadth are $7 	ext{ cm}$ and $3 	ext{ cm}$ respectively, find the height of each cuboidal vessel.

Accepted Answer

**Solution:**

1. Calculate the volume of the cylindrical vessel:
   - Radius $r = 17.5 	ext{ cm}$
   - Height $h = 18 	ext{ cm}$
   - Volume of cylinder $= \pi r^2 h = \frac{22}{7} 	imes 17.5 	imes 17.5 	imes 18$

Calculate stepwise:
   $$\frac{22}{7} 	imes 17.5 = 55$$
   $$55 	imes 17.5 = 962.5$$
   $$962.5 	imes 18 = 17325 	ext{ cm}^3$$

2. Volume of milk (80% of total volume):
   $$17325 	imes \frac{80}{100} = 13860 	ext{ cm}^3$$

3. Volume of one cuboidal vessel:
   - Length $= 7 	ext{ cm}$
   - Breadth $= 3 	ext{ cm}$
   - Height $= h$ (to be found)

4. Total volume of milk is distributed equally into 30 cuboidal vessels:
   $$30 	imes 7 	imes 3 	imes h = 13860$$

5. Calculate height $h$:
   $$30 	imes 7 	imes 3 = 630$$
   $$630 	imes h = 13860$$
   $$h = \frac{13860}{630} = 22 	ext{ cm}$$

**Answer:** $22 	ext{ cm}$

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