Question

The Diameter of a sphere is decreased by 25%. By what percent does its curved surface area decrease

Answer 1

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Answer 2

$\huge\bold\red{Question}$

The diameter of a sphere is decreased by 25% by what percent does its curved surface area decrease?

$\huge\bold\green{Solution}$

$\bold{Let\: the\: diameter \:of\: the\: sphere\: be,“d”}$

$\bold{Radius \:of \:sphere, r_1 = \frac{d}{2}}$

$\bold{New\:radius \:of\: sphere, say\: r_2}$

=$\bold{\frac{d}{2} \times (\frac{1-25}{100}) = \frac{3d}{8} }$

$\bold{Curved\: surface \:area\: of\: sphere\: (CSA)_1}$

=$\bold{4π {r_{1}}^{2} = 4π \times \frac{d}{2}^2 = πd^2 \: …eqn \: ....(i)}$

$\bold{Curved\: surface\: area \:of \:sphere\: when\: radius\: is\: decreased (CSA)_2}$

$\bold{=4π{r_{2}}^{2} = 4π×(\frac{3d}{8})^2 = (\frac{9}{16})πd^2\:....eqn(ii)}$

$\bold{From \:equation \:(i) \:and\: (ii), \:we\: have}$

$\bold{Decrease\: in \:surface\: area\: of\: sphere}$

=$\bold{(CSA)_1 – (CSA)_2}$

$\bold{= πd^2 – (\frac{9}{16})πd^2}$

$\bold{= (\frac{7}{16})πd^2}$

$\bold{Percentage \: decrease \: in \: surface \: area \: of \: sphere}$

= $\bold{\frac{(CSA)_1 - (CSA)_2}{(CSA)_1} \times 100 }$

=$\bold{(\frac{7d^2}{16d^2})}×100$

= $\bold{\frac{700}{16} }$

= $\bold{43.75\: \%}$

ᴛʜᴇʀᴇғᴏʀᴇ, ᴛʜᴇ ᴘᴇʀᴄᴇɴᴛᴀɢᴇ ᴅᴇᴄʀᴇᴀsᴇ ɪɴ ᴛʜᴇ sᴜʀғᴀᴄᴇ ᴀʀᴇᴀ ᴏғ ᴛʜᴇ sᴘʜᴇʀᴇ ɪs $43.75\:\%$