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

CBSE BoardEnglish MediumSTD 10MathsVolume and Surface Areas 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)s true.

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