The ratio of the volume of a cube to the volume of a sphere is $21:88$, and the total surface area of the cube is $294 ext{ cm}^2$. Find the curved surface area of the sphere (in $ ext{cm}^2$).

Question

The ratio of the volume of a cube to the volume of a sphere is $21:88$, and the total surface area of the cube is $294 	ext{ cm}^2$. Find the curved surface area of the sphere (in $ 	ext{cm}^2$).

Accepted Answer

**Solution:**

1. Let the side of the cube be $a$ and the radius of the sphere be $r$.

2. Given the total surface area of the cube is $294 	ext{ cm}^2$.

Total surface area of cube = $6a^2 = 294$

So, $a^2 = \frac{294}{6} = 49$

Therefore, $a = \sqrt{49} = 7 	ext{ cm}$.

3. Volume of cube = $a^3 = 7^3 = 343 	ext{ cm}^3$.

4. Let the volume of the sphere be $V_s$ and volume of cube be $V_c$.

Given ratio of volumes: $V_c : V_s = 21 : 88$

We know $V_c = 343$, so:

$$\frac{343}{V_s} = \frac{21}{88} \implies V_s = \frac{343 	imes 88}{21}$$

Simplify:

$$V_s = \frac{343 	imes 88}{21} = 343 	imes \frac{88}{21}$$

Since $343 = 7^3$ and $21 = 7 	imes 3$, simplify:

$$V_s = 7^3 	imes \frac{88}{7 	imes 3} = 7^2 	imes \frac{88}{3} = 49 	imes \frac{88}{3} = \frac{4312}{3} \approx 1437.33 	ext{ cm}^3$$

5. Volume of sphere formula:

$$V_s = \frac{4}{3} \pi r^3$$

So,

$$\frac{4}{3} \pi r^3 = \frac{4312}{3}$$

Multiply both sides by 3:

$$4 \pi r^3 = 4312$$

$$r^3 = \frac{4312}{4 \pi} = \frac{4312}{4 	imes 3.1416} = \frac{4312}{12.5664} \approx 343.04$$

6. Taking cube root:

$$r = \sqrt[3]{343.04} \approx 7 	ext{ cm}$$

7. Curved surface area of sphere (surface area) = $4 \pi r^2$

$$= 4 	imes 3.1416 	imes 7^2 = 4 	imes 3.1416 	imes 49 = 615.75 \approx 616 	ext{ cm}^2$$

**Answer:** $616 	ext{ cm}^2$

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