Question

If the ratio of volumes of two spheres is 1:8, then the ratio of their surface area is :

• A. 1:2
• B. 1:4
• C. 1:8
• D. 1:16

Hint:

If the ratio of volumes of two similar 3D shapes (like spheres) is \(V_1:V_2 = k^3\), then the ratio of their corresponding linear dimensions (like radii) is \(r_1:r_2 = k\), and the ratio of their surface areas is \(A_1:A_2 = k^2\). Given \(V_1:V_2 = 1:8\). Since \(8 = 2^3\), we have \(k^3 = (1/2)^3\) if we consider \(V_1/V_2 = 1/8\). This means the ratio of radii \(r_1:r_2 = 1:2\). (So \(k=1/2\)) Then the ratio of surface areas \(A_1:A_2 = (r_1/r_2)^2 = (1/2)^2 = 1:4\).

Answer 1

The Correct Option is B

Concept: This problem involves the formulas for the volume and surface area of a sphere.
Volume of a sphere (\(V\)) = \(\frac{4}{3}\pi r^3\), where \(r\) is the radius.
Surface Area of a sphere (\(A\)) = \(4\pi r^2\), where \(r\) is the radius. Step 1: Set up the ratio of volumes Let the two spheres have radii \(r_1\) and \(r_2\), volumes \(V_1\) and \(V_2\), and surface areas \(A_1\) and \(A_2\). We are given that the ratio of their volumes is 1:8. \[ \frac{V_1}{V_2} = \frac{1}{8} \] Substitute the volume formula: \[ \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} = \frac{1}{8} \] The term \(\frac{4}{3}\pi\) cancels out from the numerator and denominator: \[ \frac{r_1^3}{r_2^3} = \frac{1}{8} \] This can be written as: \[ \left(\frac{r_1}{r_2}\right)^3 = \frac{1}{8} \] Step 2: Find the ratio of their radii To find the ratio of the radii \(\frac{r_1}{r_2}\), take the cube root of both sides: \[ \frac{r_1}{r_2} = \sqrt[3]{\frac{1}{8}} \] Since \(\sqrt[3]{1} = 1\) and \(\sqrt[3]{8} = 2\) (because \(2^3 = 8\)), \[ \frac{r_1}{r_2} = \frac{1}{2} \] So, the ratio of their radii is 1:2. Step 3: Find the ratio of their surface areas The surface area of a sphere is \(A = 4\pi r^2\). We need to find the ratio \(\frac{A_1}{A_2}\): \[ \frac{A_1}{A_2} = \frac{4\pi r_1^2}{4\pi r_2^2} \] The term \(4\pi\) cancels out: \[ \frac{A_1}{A_2} = \frac{r_1^2}{r_2^2} \] This can be written as: \[ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 \] Step 4: Substitute the ratio of radii From Step 2, we found \(\frac{r_1}{r_2} = \frac{1}{2}\). Substitute this into the surface area ratio: \[ \frac{A_1}{A_2} = \left(\frac{1}{2}\right)^2 \] \[ \frac{A_1}{A_2} = \frac{1^2}{2^2} = \frac{1}{4} \] So, the ratio of their surface areas is 1:4.