Geometry & Mensuration
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Q.

Given five concentric squares. If the area of the circle inside the smallest square is 77 square units and the distance between the corresponding corners of consecutive squares is 1.5 units, find the difference in the areas of the outermost and innermost squares.

 A.

1254 sq units

 B.

1008 sq units

 C.

877 sq units

 D.

240 sq units

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Solution:
Option(D) is correct

Here we see that diameter of the circle is equal to the side of the innermost square that is,

\(\pi r^2=77\)

\(r=3.5\sqrt{2}\)

\(2r=7\sqrt{2}\)

Then the diagonal of the square is 14 sq units.

Which means the diagonal of the fifth sqaure would be 14+12 units = 26.

Which means the side of the fifth square would be $\dfrac{26}{\sqrt{2}}$

Therefore, the area of the fifth sqaure $= 338$ sq units.

Area of the first square $= 98$ sq units.

Hence, the difference would be $240 (=338-98)$ sq units.

Edit: Thank you anubhav goel for pointing out the mistake. Solution has been updated.

Edit2: Thank you Manoj, changed the length of side of fifth square from $26\sqrt{2}$ to $\frac{26}{\sqrt{2}}$ and hence the final answer


(7) Comment(s)


Ashwin
 ()

How that 12 units came?

Which means the diagonal of the fifth sqaure would be 14+12 units = 26



Manoj
 ()

From side 7√2, diagonal is 14 ie you have multiplied √2. But from diagonal of fifth square ie 26, you should have divided √2 and not multiplied to get the side. So side of the fifth square should have been 13√2. Correct me if I am wrong.


Deepak
 ()

You are right manoj and side of the fifth square should be $\dfrac{26}{\sqrt{2}}$ indeed. Changed the option choices and final answer.


Pawan Sharma
 ()

How could the diameter of the inner circle be 14 units?

Please explain.

If i am not wrong inner diagonal should be $7\sqrt{2}$.


Deepak
 ()

Hey Pawan,

Diameter of the CIRCLE is $7\sqrt{2}$ only. It's the DIAGONAL OF THE SQUARE which is 14 sq. units.

To give you calculations for calculating the diagonal of the square, $D$,

$=\sqrt{(7\sqrt{2})^2+(7\sqrt{2})^2}$

$=\sqrt{(49\times 2)+(49\times 2)}$

$=\sqrt{196}$

$=\textbf{14 units}$


Anubhav Goel
 ()

i think your aproach is wrong, shouldn't the diameter of circle equal to side of the innermost square?


Deepak
 ()

You are right Anubhav, there was a typo in the question. Calculations were correct but some words got jumbled up. Updated the solution.