The volume of a sphere is $2145 \frac{11}{21}$ cm$^3$. Its radius is equal to

### Concept & Volume of a Sphere To find the radius of a sphere when its volume is given, use the standard volume formula and solve for $r$. $$ V = \frac{4}{3} \pi r^3 $$ ### Step-by-Step Solution - The given volume is a mixed fraction: $2145 \frac{11}{21}$. - First, convert this into an improper fraction: $\frac{2145 \times 21 + 11}{21} = \frac{45045 + 11}{21} = \frac{45056}{21}$. - Equate this to the volume formula: $\frac{4}{3} \times \frac{22}{7} \times r^3 = \frac{45056}{21}$. - Notice that the denominator on the left is $3 \times 7 = 21$, which perfectly matches the denominator on the right. - The equation simplifies to: $4 \times 22 \times r^3 = 45056$. - $88 \times r^3 = 45056$. - Isolate $r^3$: $r^3 = \frac{45056}{88}$. - Perform the division: $45056 \div 8 = 5632$. Then $5632 \div 11 = 512$. - So, $r^3 = 512$. - Take the cube root of both sides: $r = \sqrt[3]{512} = 8$ cm. ### Exam Strategy & Shortcut When dividing $45056$ by $88$, look at the options. The options are 7, 8, and 9. $7^3 = 343 \implies 343 \times 88 \approx 30000$ (Too small). $9^3 = 729 \implies 729 \times 88 \approx 64000$ (Too large). $8^3 = 512 \implies 512 \times 88 = 45056$. This verifies the answer immediately without complex long division. ### Common Pitfall Making an arithmetic error when converting the large mixed number $2145 \frac{11}{21}$ into an improper fraction. Take time to multiply $2145 \times 21$ accurately. ### Final Answer Therefore, the correct answer is **8 cm**.