The total surface area of a cylinder is 6512 cm² and the circumference of its base is 88 cm. Find :

(i) its radius

(ii) its volume

(i) Given:

The total surface area of the cylinder is 6512 cm².

The circumference of its base is 88 cm

Circumference of circle = 2πr

$⇒ 2 \times \dfrac{22}{7} \times r = 88\\[1em] ⇒ \dfrac{44}{7} \times r = 88\\[1em] ⇒ r = \dfrac{88 \times 7}{44}\\[1em] ⇒ r = \dfrac{616}{44}\\[1em] ⇒ r = 14$

Hence, the radius of the cylinder is 14 cm.

(ii) Total surface area of cylinder = 6512 cm²

$⇒ 2πr(r + h) = 6512 cm^2\\[1em] ⇒ 2 \times \dfrac{22}{7} \times 14 \times (14 + h) = 6512\\[1em] ⇒ \dfrac{44}{7} \times 14 \times (14 + h) = 6512\\[1em] ⇒ \dfrac{616}{7}\times (14 + h) = 6512\\[1em] ⇒ 88 \times (14 + h) = 6512\\[1em] ⇒ (14 + h) = \dfrac{6512}{88}\\[1em] ⇒ (14 + h) = 74\\[1em] ⇒ h = 74 - 14\\[1em] ⇒ h = 60$

Height of the cylinder = 60 cm

Volume of the cylinder = πr²h

$= \dfrac{22}{7} \times 14^2 \times 60\\[1em] = \dfrac{22}{7} \times 196 \times 60\\[1em] = \dfrac{258,720}{7}\\[1em] = 36,960$

Hence, the volume of the cylinder is 36,960 cm³.