The respective ratios between the numerical values of a curved surface area

Step 1: Let the radius and height be r and h.

• CSA = $2 \pi r h$

• Volume = $\pi r^2 h$

Given CSA : Volume = 1 : 7 → $2 \pi r h : \pi r^2 h = 1 : 7$

Step 2: Simplify the ratio

$\frac{2 \pi r h}{\pi r^2 h} = \frac{2}{r} = \frac{1}{7}$

So $r = 14$ units

Step 3: Use diameter : height = 7 : 6

Diameter $= 2r = 28$ → $28 : h = 7 : 6$

Solve for $h$: $h = 28 \cdot 6 / 7 = 24$ units

Step 4: Total surface area (TSA) of cylinder

TSA = CSA + 2 × base area = $2 \pi r h + 2 \pi r^2 = 2 \pi r (h + r)$

Substitute $r = 14$, $h = 24$, $\pi = 22/7$:

TSA = $2 \cdot (22/7) \cdot 14 \cdot (24 + 14) = 2 \cdot (22/7) \cdot 14 \cdot 38$

Step 5: Simplify

$2 \cdot (22/7) \cdot 14 = 2 \cdot 22 \cdot 2 = 88$

Multiply by 38: $88 \cdot 38 = 3344$

Total surface area = 3344 unit²