The sum of the radius of the base and the height of a solid cylinder is 37 metres. If the total surface area of the cylinder be 1628 sq. metres, its volume is

### Concept & Algebraic Substitution The problem gives the sum of radius and height, which perfectly matches a component of the Total Surface Area (TSA) formula. Total Surface Area: $$TSA = 2\pi r(r + h)$$ Volume: $$V = \pi r^2 h$$ ### Step-by-Step Solution * **Given:** $(r + h) = 37$ m, TSA = 1628 m$^2$. * Use the TSA formula to find $r$: $$2\pi r(r + h) = 1628$$ $$2 \times \frac{22}{7} \times r \times 37 = 1628$$ $$\frac{44}{7} \times r \times 37 = 1628$$ * Isolate $r$: $$r = \frac{1628 \times 7}{44 \times 37}$$ Notice that $44 \times 37 = 1628$. So, $$r = \frac{1628 \times 7}{1628} = 7 \text{ m}$$ * Find $h$ using the sum equation: $$7 + h = 37 \Rightarrow h = 30 \text{ m}$$ * Calculate the volume: $$V = \frac{22}{7} \times 7^2 \times 30$$ $$V = 22 \times 7 \times 30 = 154 \times 30 = 4620 \text{ m}^3$$ ### Exam Strategy & Shortcut Look for cancellation opportunities. When you see $2 \times 22 \times 37 \times r / 7 = 1628$, test if 1628 is divisible by 37 (it ends in 8, $7 \times 4 = 28$, so try $37 \times 44 = 1628$). The calculation collapses immediately to $r = 7$. ### Common Pitfall A common pitfall is expanding $2\pi r(r+h)$ into $2\pi r^2 + 2\pi rh = 1628$ and attempting to solve a quadratic equation, completely missing the fact that $(r+h)$ is given as a single constant value. ### Final Answer Therefore, the correct answer is **4620 m$^3**.