Question

What is the ratio for the volumes of two similar spheres, given that the ratio of their radii is 5:6?

Answer 1

Answer:

125 : 216

Step-by-step explanation:

Let the volumes of the spheres be V₁ and V₂ respectively.

Volume (Sphere radius = 5) :

• V₁ = 4/3π(5)³
• V₁ = 4/3π x 125
• V₁ = 500/3π

Volume (Sphere radius = 6) :

• V₂ = 4/3π(6)³
• V₂ = 4/3π x 216
• V₂ = 864/3π

Taking the ratio of V₁ and V₂ :

• V₁/V₂
• 500/3π x 3π/864
• 500/864
• 125 : 216

Answer 2

Given ratio: $(\text{Radius of Sphere}_1 : \text{Radius of Sphere}_2) = (5 :6)$

Step-1) Multiply both sides of the ratio by x:

    $\implies (\text{Radius of Sphere}_1 : \text{Radius of Sphere}_2) = (5x :6x)$

    $\implies \text{Radius of Sphere}_1 = 5x;\ \text{Radius of Sphere}_2 = 6x$

Step-2) Substitute the radiuses of each sphere into the "Volume formula"

               Volume of sphere₁                                Volume of sphere₂      

    $\implies \ \ \ \ \ \ \ \ \dfrac{4}{3} \pi r^{3} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \dfrac{4}{3} \pi r^{3}$

    $\implies \ \ \ \ \ \ \ \ \dfrac{4}{3} \pi (5x)^{3} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \dfrac{4}{3} \pi (6x)^{3}$

    $\implies \ \ \ \ \ \ \ \ \dfrac{4}{3} \pi (5x)(5x)(5x) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \dfrac{4}{3} \pi (6x)(6x)(6x)$

    $\implies \ \ \ \ \ \ \ \ \dfrac{4}{3} \pi (125x^{3} ) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4 \pi (2x)(6x)(6x)$

    $\implies \ \ \ \ \ \ \ \ \dfrac{500x^{3} }{3} \pi \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 288x^{3} \pi$

Step-3) Plug the volume of both spheres in ratio form:

The ratio for the volumes of the two spheres must be in the respective place of the ratio for the radiuses of the two similar spheres. Therefore,

Ratio form: $\underline{(\small\text{Volume of sphere}_1 : \text{Volume of sphere}_2)}$

    $\implies \dfrac{500x^{3} }{3} \pi: 288x^{3} \pi$

Step-4) Simplify both sides of the ratio:

    $\implies \dfrac{500x^{3} }{3}: 288x^{3}$

    $\implies \dfrac{500 }{3}: 288$

    $\implies \dfrac{500 }{3}} \div{\dfrac{288}{1} } \implies \dfrac{500}{3} \times \dfrac{1}{288} \implies \dfrac{500}{864} \implies \boxed{\dfrac{125}{216}}$

Therefore, the ratio for the volumes of two similar spheres is 125 : 216.