If the circumference of the base of a cylinder is 88 cm, and the surface area and height is 24 cm, then find: (a) Area of the base. (b) Curved surface area sum of radius and height is 24. (c) Total surface area (d) Volume

(a) r = 7 cm, h = 10 cm

Step 1: Curved surface area (CSA) = $2\pi rh$

Substitute the values:
$2\pi \times 7 \times 10 = 140\pi$ cm²

Step 2: Total surface area (TSA) = $2\pi r(h + r)$
$2\pi \times 7 \times (10 + 7) = 2\pi \times 7 \times 17 = 238\pi$ cm²

Step 3: Volume (V) = $\pi r^2 h$
$\pi \times 7^2 \times 10 = \pi \times 49 \times 10 = 490\pi$ cm³

140\pi cm², 238\pi cm², 490\pi cm³

(b) r = 2.5 cm = $\dfrac{5}{2}$ cm, h = 7 cm

Step 1: CSA = $2\pi rh$
$2\pi \times \dfrac{5}{2} \times 7 = 2\pi \times \dfrac{35}{2} = 35\pi$ cm²

Step 2: TSA = $2\pi r(h + r)$
$h + r = 7 + \dfrac{5}{2} = \dfrac{19}{2}$
$2\pi \times \dfrac{5}{2} \times \dfrac{19}{2} = 2\pi \times \dfrac{95}{4} = \dfrac{95\pi}{2}$ cm²

Step 3: V = $\pi r^2 h$
$r^2 = \left(\dfrac{5}{2}\right)^2 = \dfrac{25}{4}$
$\pi \times \dfrac{25}{4} \times 7 = \pi \times \dfrac{175}{4} = \dfrac{175\pi}{4}$ cm³

35\pi cm², \dfrac{95\pi{2} cm², \dfrac{175\pi}{4} cm³}

(c) Circumference of base = 14 cm, h = 10 cm

Step 1: Circumference = $2\pi r = 14$
$r = \dfrac{14}{2\pi} = \dfrac{7}{\pi}$ cm

Step 2: TSA = $2\pi r(h + r)$
$h + r = 10 + \dfrac{7}{\pi}$
$2\pi \times \dfrac{7}{\pi} \times \left(10 + \dfrac{7}{\pi}\right) = 14 \times \left(10 + \dfrac{7}{\pi}\right) = 140 + \dfrac{98}{\pi}$ cm²

r = \dfrac{7{\pi} cm, TSA = 140 + \dfrac{98}{\pi} cm²}

(d) r + h = 24 cm, h = 21 cm (from diagram)

Step 1: r = 24 - 21 = 3 cm

Step 2: CSA = $2\pi rh$
$2\pi \times 3 \times 21 = 126\pi$ cm²

Step 3: TSA = $2\pi r(h + r)$
$h + r = 21 + 3 = 24$
$2\pi \times 3 \times 24 = 144\pi$ cm²

CSA = 126\pi cm², TSA = 144\pi cm²