$1496$ cm³ of metal is used to cast a pipe of length $28$ cm. If the internal radius of the pipe is $8$ cm, the outer radius of the pipe is

### Concept & Finding Unknown Radius When the volume, length, and one radius of a hollow cylinder are known, we can isolate the unknown radius using the volume formula. $$ V = \pi \times h \times (R^2 - r^2) $$ ### Step-by-Step Solution - **Given:** - Volume $V = 1496$ cm$^3$. - Length $h = 28$ cm. - Inner radius $r = 8$ cm. - Let the outer radius be $R$. - **Calculation:** - Substitute the known values into the formula: - $1496 = \frac{22}{7} \times 28 \times (R^2 - 8^2)$ - Simplify the constants: - $1496 = 22 \times 4 \times (R^2 - 64)$ - $1496 = 88 \times (R^2 - 64)$ - Isolate the radius term: - $R^2 - 64 = \frac{1496}{88}$ - $R^2 - 64 = 17$ - $R^2 = 17 + 64 = 81$ - $R = \sqrt{81} = 9$ cm. ### Exam Strategy & Shortcut Notice that $1496 \div 88$ must end in $7$ because $8 \times 7 = 56$. This makes division faster: $88 \times 10 = 880$, and the rest follows naturally to yield $17$. ### Common Pitfall Forgetting to take the square root at the very end after isolating $R^2$. ### Final Answer Therefore, the correct answer is **9 cm**.