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Sequences and Series — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsSequences and Series1 Mark
MCQ
Choose the correct answer. The lengths of three unequal edges of a rectangular solid block are in $G.P.$ If the volume of the block is $216\ cm^3$ and the total surface area is $252\ cm^2$, then the length of the longest edge is:
  • ✓
    $12\ cm$
  • B
    $6\ cm$
  • C
    $18\ cm$
  • D
    $3\ cm$
✓

Answer

Correct option: A.
$12\ cm$
Let the length, breadth and height of rectangular solid block be $\frac{\text{a}}{\text{r}}, a$ and $ar,$ respectively.
$\therefore\ \text{Volume}=\frac{\text{a}}{\text{r}}\times\text{a}\times\text{ar}=216\text{cm}^3$
$\Rightarrow\text{a}^3=216=6^3$
$\Rightarrow\text{a}=6$
Also, Surface area $=2\Big(\frac{\text{a}}{\text{r}}.\text{a}+\text{a}.\text{ar}+\frac{\text{a}}{\text{r}}.\text{ar}\Big)=252$
$\Rightarrow2\text{a}^2\Big(\frac{1}{\text{r}}+\text{r}+1\Big)=252$
$\Rightarrow2\times36\Big(\frac{1+\text{r}^2+\text{r}}{\text{r}}\Big)=252$
$\Rightarrow2(1+\text{r}^2+\text{r})=7\text{r}$
$\Rightarrow2\text{r}^2-5\text{r}+2=0$
$\Rightarrow(2\text{r}-1)(\text{r}-2)=0$
$\therefore\ \text{r}=\frac{1}{2},2$
For $\text{r}=\frac{1}{2}:$ Length $=\frac{\text{a}}{\text{r}}=\frac{6\times2}{1}=12,$ Breadth $= a = 6$
Height $=\text{ar}=6\times\frac{1}{2}=3$
For $r = 2:$ Length $=\frac{\text{a}}{\text{r}}=\frac{6}{2}=3,$ Breadth $= a = 6$
Height $= ar = 6 \times 2 = 12$

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