The total surface area of cylinder is 352 cm². If its height is 10 cm, then find the diameter of the base.

Given, Total surface area = 352 cm² and height = 10 cm.

We know that Total surface area = 2πr(h + r).

∴ 2πr(h + r) = 352

Putting values we get,

$\Rightarrow 2 \times \dfrac{22}{7} \times r \times (10 + r) = 352 \\[1em] \Rightarrow \dfrac{2 \times 22 \times r \times (10 + r)}{7} = 352 \\[1em] \Rightarrow 44r(10 + r) = 352 \times 7 \\[1em] \Rightarrow 440r + 44r^2 = 2464 \\[1em]$

Dividing the complete equation by 44,

$\Rightarrow 10r + r^2 = 56 \\[1em] \Rightarrow r^2 + 10r - 56 = 0 \\[1em] \Rightarrow r^2 + 14r - 4r - 56 = 0 \\[1em] \Rightarrow r(r + 14) - 4(r + 14) = 0 \\[1em] \Rightarrow (r - 4)(r + 14) = 0 \\[1em] \Rightarrow r - 4 = 0 \text{ or } r + 14 = 0 \\[1em] \Rightarrow r = 4 \text{ or } r = -14.$

Since radius cannot be negative hence, r ≠ -14.

∴ r = 4 cm.

Diameter = r × 2 = 4 × 2 = 8 cm.

Hence, the diameter of the base = 8 cm.