A man makes complete use of $405 \mathrm{cc}$ of iron, $783 \mathrm{cc}$ of aluminium, and $351 \mathrm{cc}$ of copper to make a number of solid right circular cylinders of each type of metal. These cylinders have the same volume and each of these has radius $3 \mathrm{~cm}$. If the total number of cylinders is to be kept at a minimum, then the total surface area of all these cylinders, in sq cm, is

we need to make cylinders from three different metals using all the available material. The key insight is that to minimize the total number of cylinders, we want to make each cylinder as large as possible while keeping them all the same size.

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Since we must use all the material and all cylinders have the same volume, each cylinder's volume must be a common divisor of all three quantities (405, 783, 351). To minimize the number of cylinders, we need the largest possible common divisor, which is the HCF (Highest Common Factor).

Finding HCF using prime factorization:

405 = 3⁴ × 5 = 81 × 5 = 27 × 15

783 = 3³ × 29 = 27 × 29

351 = 3³ × 13 = 27 × 13

HCF = 27

Therefore, each cylinder uses 27 cc of material.

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we know:

Volume of each cylinder = 27 cc

Radius = 3 cm

Volume formula: V = πr²h

$27 = π × 3² × h$

$27 = 9πh$

$h = \dfrac{27}{9π} = \dfrac{3}{π}$ cm

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Total material available = 405 + 783 + 351 = 1539 cc

Since each cylinder uses 27 cc:

Number of cylinders = $\dfrac{1539}{27} = 57$

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Surface area of one cylinder:

$\text{Surface Area} = 2πrh + 2πr² = 2πr(h + r)$

$= 2π × 3 × \left(\dfrac{3}{π} + 3\right)$

$= 6π × \left(\dfrac{3}{π} + 3\right)$

$= 6π × \dfrac{3 + 3π}{π}$

$= 6 × (3 + 3π)$

$= 18(1 + π)$

Total surface area of all 57 cylinders:

$= 57 × 18(1 + π)$

$= 1026(π + 1)$ cm²

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When you need to minimize the number of identical objects made from different quantities of material, always find the HCF of those quantities. This gives you the maximum possible size for each object, thus minimizing the total count.

Answer: 1026(π + 1) cm²