# Twenty-seven solid iron spheres, each of radius *r *and surface area S are melted to form a sphere with surface area S′. Find the

(i) radius *r*′ of the new sphere, (ii) ratio of S and S′

**Solution:**

Since 27 solid iron spheres are melted to form a single solid sphere, the volume of the newly formed sphere will be equal to the volume of 27 solid iron spheres together.

The surface area of a sphere = 4πr^{2}

The volume of a sphere = 4/3πr^{3}

Therefore, The volume of 27 solid spheres with radius r = 27 × [4/3πr^{3}] = 36πr^{3} ---------------- (1)

Also, the volume of the new sphere with radius r' = 4/3πr'^{3}----------------- (2)

(i) Volume of the new sphere = Volume of 27 solid spheres

(4/3) πr'^{3} = 36πr^{3} [From equation (1) and (2)]

⇒ r'^{3} = 36πr³ × 3/4π

⇒ r'^{3} = 27r³

⇒ r' = ∛27r³

r' = 3r

Radius of the new sphere, r' = 3r

(ii) Ratio of S and S′

Now, surface area of each iron sphere, S = 4πr^{2}

Surface area of the new sphere, S' = 4πr'^{2} = 4π(3r)^{2} = 36πr^{2}

The ratio of the S and S’ = 4πr^{2}/36πr^{2} = 1/9

Hence, the radius of new sphere is 3r and the ratio of S and S’ is 1:9.

**Video Solution:**

## Twenty-seven solid iron spheres, each of radius *r *and surface area S are melted to form a sphere with surface area S′. Find the i) radius *r*′ of the new sphere, ii) ratio of S and S′

### NCERT Solutions for Class 9 Maths - Chapter 13 Exercise 13.8 Question 9:

**Summary:**

It is given that twenty-seven solid iron spheres, each of radius *r *and surface area S are melted to form a sphere with surface area S′. We have found that the radius r of the new sphere is 3r and the ratio of S and S’ is 1:9