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Volume and Surface area of Solids — Maths STD 10 — Question

Maharashtra BoardEnglish MediumSTD 10MathsVolume and Surface area of Solids1 Mark
MCQ
Assertion and Reason Type
Each question consists of two statements, namely, Assertion $(A)$ and Reason $(R).$ For selecting the correct answer, use the following code:
Assertion $(A)$
Reason $(R)$
If the volumes of two spheres are in the ratio $27 : 8$ then their surface areas are in the ratio $3: 2.$
Volume of a sphere $=\frac{4}{3}\pi\text{R}^3.$
Surface area of a sphere $=4\pi\text{R}^2.$
  • A
    Both Assertion $(A)$ and Reason $(R)$ are true and Reason $(R)$ is a correct explanation of Assertion $(A).$
  • B
    Both Assertion $(A)$ and Reason $(R)$ are true but Reason $(R)$ is not a correct explanation of Assertion $(A).$
  • C
    Assertion $(A)$ is true and Reason $(R)$ is false.
  • ✓
    Assertion $(A)$ is false and Reason $(R)$ is true.
✓

Answer

Correct option: D.
Assertion $(A)$ is false and Reason $(R)$ is true.
Let r and R be the radii of the two sheres.
Ratio of their volumes $=\frac{27}{8}$
$\Rightarrow\frac{\frac{4}{3}\pi\text{r}^3}{\frac{4}{3}\pi\text{R}^3}=\frac{27}{8}$
$\Rightarrow\frac{\text{r}^3}{\text{R}^3}=\frac{27}{8}$
$\Rightarrow\frac{\text{r}}{\text{R}}=\frac{3}{2}$
Ratio of their surface areas $=\frac{4\pi\text{r}^2}{4\pi\text{R}^2}$
$=\Big(\frac{\text{r}}{\text{R}}\Big)^2$
$=\Big(\frac{3}{2}\Big)^2$
$=\frac{9}{4}$
So, the Assertion $(A)$ is false.
The reason $(R)$ is true.

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