Question

Three spheres whose radii are in the ratio 3 : 2 : 1 are melted and converted into a single cube whose diagonal is 15 cm. Find the radii of the three spheres.​

Answer 1

» Question :

Three Spheres whose Radii are in the ratio 3 : 2 : 1 are melted and converted into a single Cube whose diagonal is 15 cm. Find the Radii of the three Spheres.

» To Find :

The Radii of the three Spheres .

» Given :

• Ratio of the radii of the three Spheres = 3 : 2 : 1

• Diagonal of the Sphere = 15 cm

» We Know :

Diagonal of a Cube :

$\sf{\underline{\boxed{D_{c} = \sqrt{3}a}}}$

Volume of a Cube :

$\sf{\underline{\boxed{V_{c} = a^{3}}}}$

Volume of a Sphere :

$\sf{\underline{\boxed{V_{s} = \dfrac{4}{3}\pi r^{3}}}}$

» Concept :

According to the question ,it says that the Cube was formed by recasting the three Spheres .

So the Volume of the three Spheres will be Equal to the Volume of the cube.i.e,

$\sf{V_{Sphere} = V_{cube}}$

But first ,we have to find the edge of the cube , so that we can find the Volume of the cube.

The edge can be find using the formula for Diagonal of a Cube.

• Diagonal = 15 cm

Formula :

$\sf{\underline{\boxed{D_{c} = \sqrt{3}a}}}$

Substituting the values in it ,we get :

$\sf{\Rightarrow 15 = \sqrt{3}a}$

$\sf{\Rightarrow \dfrac{15}{\sqrt{3}} = a}$

Hence ,the edge of the Cube is $\dfrac{15}{\sqrt{3}}$ cm.

Now ,by this we can find the Volume of the Cube.

» Solution :

Volume of the Cube :

• Edge = $\dfrac{15}{\sqrt{3}}$ cm

Formula :

$\sf{\underline{\boxed{V_{c} = a^{3}}}}$

By Substituting the value in it ,we get :

$\sf{\Rightarrow V_{c} = \dfrac{15}{\sqrt{3}}^{3}}$

$\sf{\Rightarrow V_{c} = s}$

$\sf{\Rightarrow V_{c} = 649.5(approx.) cm^{3}}$

Hence ,the Volume of the Cube is 649.5 cm³.

Radii of the Three Spheres :

Given that the , Volume of the Cube is Volume of the Sphere.

Let the three radii be 3x , 2x and 1x.

So , we get :

$\sf{V_{S} = V_{c}}$

$\sf{\Rightarrow \dfrac{4}{3}\pi r_{1}^{3} + \dfrac{4}{3}\pi r_{2}^{3} + \dfrac{4}{3}\pi r_{3}^{3} = 649.5}$

$\sf{\Rightarrow \dfrac{4}{3}\pi(r_{1}^{3} + r_2^{3} + r_{3}^{3}) = 649.5}$

$\sf{\Rightarrow \dfrac{4}{3}\pi((3x)^{3} + (2x)^{3} + (x)^{3}) = 649.5}$

$\sf{\Rightarrow \dfrac{4}{3}\pi(27x^{3} + 8x^{3} + x^{3}) = 649.5}$

$\sf{\Rightarrow \dfrac{4}{3} \times \dfrac{22}{7} \times 36x^{3} = 649.5}$

$\sf{\Rightarrow \dfrac{4}{3} \times 22 \times 5.14x^{3} = 649.5}$

$\sf{\Rightarrow \dfrac{452.32}{3} \times x^{3} = 649.5}$

$\sf{\Rightarrow \dfrac{452.32}{3}x^{3} = 649.5}$

$\sf{\Rightarrow 452.32x^{3} = 649.5 \times 3}$

$\sf{\Rightarrow 452.32x^{3} = }$

$\sf{\Rightarrow x^{3} = \dfrac{1948.5}{452.32}}$

$\sf{\Rightarrow x^{3} = \cancel{\dfrac{1948.5}{452.32}}}$

$\sf{\Rightarrow x^{3} = 4.3(approx.)}$

$\sf{\Rightarrow x = \sqrt[3]{4.3}}$

$\sf{\Rightarrow x = 1.6(approx.) cm}$

Hence ,the value of x is 1.6 cm

So ,The radii of the spheres are ,

• 3x ➝ 3 × 1.6 ➝ 4.8 cm
• 2x ➝ 2 × 1.6 ➝ 3.2 cm
• x ➝ 1.6 cm

Hence , the radii of the spheres are 4.8 cm , 3.2 cm and 1.6 cm

» Additional information :

• Volume of a Cylinder = πr²h

• Volume of Cuboid = lbh

• Surface area of a Cylinder = 2πr(h + r)

• Curved surface area of a Cylinder = 2πrh