Find the number of solid spheres, each of diameter 6 cm, that can be m

1.	Find the number of solid spheres, each of diameter 6 cm, that can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm?
Answer» Given that the diameter of the sphere is 2R = 6 cm. ∴ R = \(\frac{6}{2}\) = 3 cm. Hence, the radius of the sphere is R = 3 cm. Height and diameter of cylinder are h = 45 cm and 2r = 4 cm. Hence, the radius of the cylinder is r = \(\frac{4}{2}\) = 2 cm. Therefore, the volume of the cylinder is v = \(\pi r\) ²ℎ = \(\pi\) × 2 × 2 × 45 = 180\(\pi\) cm³. And volume of the sphere V = \(\frac{4}{3}\) \(\pi r\) ³ = \(\frac{4}{3}\) \(\pi\) × 3 × 3 × 3 = 4 × 9\(\pi\) = 36\(\pi\) cm³ . Let n solid spheres are required to making a solid metal cylinder. Therefore, the volume of cylinder = n × The volume of sphere. ∴ n× 36\(\pi\) = 180\(\pi\) ⇒ n = \(\frac{180 \pi}{36\pi}\) = \(\frac{180}{36}\) = \(\frac{30}{6}\) = 5. Hence, total 5 solid spheres are required to melting for making a solid metal cylinder of given dimensions.

Find the number of solid spheres, each of diameter 6 cm, that can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm?

Answer»

Given that the diameter of the sphere is 2R = 6 cm.

∴ R = \(\frac{6}{2}\) = 3 cm.

Hence, the radius of the sphere is R = 3 cm.

Height and diameter of cylinder are h = 45 cm and 2r = 4 cm.

Hence, the radius of the cylinder is r = \(\frac{4}{2}\) = 2 cm.

Therefore, the volume of the cylinder is v = \(\pi r\) ²ℎ = \(\pi\) × 2 × 2 × 45 = 180\(\pi\) cm³.

And volume of the sphere V = \(\frac{4}{3}\) \(\pi r\) ³ = \(\frac{4}{3}\) \(\pi\) × 3 × 3 × 3 = 4 × 9\(\pi\) = 36\(\pi\) cm³ .

Let n solid spheres are required to making a solid metal cylinder.

Therefore, the volume of cylinder = n × The volume of sphere.

∴ n× 36\(\pi\) = 180\(\pi\)

⇒ n = \(\frac{180 \pi}{36\pi}\) = \(\frac{180}{36}\)

= \(\frac{30}{6}\) = 5.

Hence, total 5 solid spheres are required to melting for making a solid metal cylinder of given dimensions.