Geometry Questions and Answers

Let $x$ be the radius of the circle
$begin{aligned} therefore x^2 &= (4 -1 )^2 + (6 - 2)^2 &= 3^2 + 4^2 &= 25 &=5^2 end{aligned}$
Hence area of the circle = $pi.x^2 = 25pi$

Side of the square

$=sqrt{ 6.4 times 2.5}$$=\sqrt{6.4\times 2.5}$
$=sqrt{frac { 64 times 2.5}{100}} = sqrt{16} = 4 text m$

$text { cost} = text {Perimeter} times 15 text{paise}$

$=4 times text {side} times 15 text {paise}$$=4\times \text{side}\times 15\text{paise}$

$=4 times sqrt{625 times 10000} times 15 text {paise} = text Rs. 1500$

The volume of the metallic shell

$= frac{4}{3} pi (2)^3 - frac{4}{3} pi(1)^3 = frac{28pi}{3}$

But by question,

Volume of sphere $= frac{28pi}{3}$

$Rightarrow frac{4}{3}pi r^3 = frac{28pi}{3} Rightarrow r^3 = 7 Rightarrow r = 7^frac{1}{3}$

$text{ Volume of the cone } = frac{1}{3} times frac{22}{7} times 10^2 times 5$

$= frac{11000}{21}$

$text{ Volume of sphere } = frac{4}{3}pi times 125$

$= frac{11000}{21}$

For the bigger cylinder,
$text{ H } = 10 text{ cm }, 2 pi r = 8$

$therefore text{ V } = pi r^2 quadtext{ H } = pi (frac{16}{pi^2}) times 10 = frac{160}{pi} text{cu cm}$

Area of rhombus

$= frac{1}{2} times text{ product of diagonal }$

$= frac{1}{2} times 24 times 10$

$Big[ because text{ small diagonal } = 2sqrt{13^2 - 12^2 } = 2 times 5 = 10 Big]$

$= 120 text{ sq. m }$

$text{Volume of cylinder } = pi r^2h$

$= 1100 = pi times 5 times 5 times h$

$Rightarrow h = frac{1100}{pi times 25} text{m} = 14text{m.}$

$therefore text {Are of curved surface} = 2pi rh$

$= 2pi times 5 times 14 = 440 text{cm}^2$

Let side of square be $x$ and that of triangle is $y$

$begin{aligned} therefore 4x &= 3y x &= frac{3}{4}y text{Area of triangle } &= frac{sqrt3}{4} times y^2 16sqrt3 &= frac{sqrt3}{4} times y^2 y &= 8 text{ and } x &= frac{3}{4} times 8 = 6 text{ cm} end{aligned}$

Total surface are of cone $= pi rl + pi r^2$
$= pi 3(3 + 5) = 24pi text{ cm}^2$