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If the diameter of a sphere is decreased by 25% by what percent does its curved surface area decrease?
$
\left( a \right)43.75\% \\
\left( b \right)21.88\% \\
\left( c \right)50\% \\
\left( d \right)25\% \\
$
Answer
![Verified]()
Verified
Verified
Hint- Use formula of curved surface area of sphere $C.S.A = 4\pi {r^2}$ and also in diameter form like $C.S.A = \pi {d^2}$, d is diameter of sphere.
Let the diameter of the sphere be d.
So, the radius of the sphere is $r = \dfrac{d}{2}$ .
Curved surface area of sphere $ = 4\pi {\left( r \right)^2} = 4\pi {\left( {\dfrac{d}{2}} \right)^2}$
Curved surface area of sphere $ = \pi {d^2}$
The diameter of the sphere decreased by 25%. It means $\dfrac{d}{4}$ subtract from total diameter (d).
So, new diameter D $ = d - \dfrac{d}{4} = \dfrac{{3d}}{4}$
New curved surface area of sphere $ = \pi {D^2}$
$
\Rightarrow \pi {\left( {\dfrac{{3d}}{4}} \right)^2} \\
\Rightarrow \dfrac{{9\pi {d^2}}}{{16}} \\
$
Percentage decrease in C.S.A $ = \dfrac{{\pi {d^2} - \pi {D^2}}}{{\pi {d^2}}} \times 100$
$
\Rightarrow \dfrac{{\pi {d^2} - \dfrac{{9\pi {d^2}}}{{16}}}}{{\pi {d^2}}} \times 100 \\
\Rightarrow \dfrac{{1 - \dfrac{9}{{16}}}}{1} \times 100 \\
\Rightarrow \dfrac{7}{{16}} \times 100 \\
\Rightarrow 43.75\% \\
$
So, the correct option is (a).
Note-Whenever we face such types of problems we use some important points. Like we know decrement in diameter so we use diameter in the formula of curved surface area of sphere rather than radius then after finding new curved surface area we can easily get percentage decreases in curved surface area.
Let the diameter of the sphere be d.
So, the radius of the sphere is $r = \dfrac{d}{2}$ .
Curved surface area of sphere $ = 4\pi {\left( r \right)^2} = 4\pi {\left( {\dfrac{d}{2}} \right)^2}$
Curved surface area of sphere $ = \pi {d^2}$
The diameter of the sphere decreased by 25%. It means $\dfrac{d}{4}$ subtract from total diameter (d).
So, new diameter D $ = d - \dfrac{d}{4} = \dfrac{{3d}}{4}$
New curved surface area of sphere $ = \pi {D^2}$
$
\Rightarrow \pi {\left( {\dfrac{{3d}}{4}} \right)^2} \\
\Rightarrow \dfrac{{9\pi {d^2}}}{{16}} \\
$
Percentage decrease in C.S.A $ = \dfrac{{\pi {d^2} - \pi {D^2}}}{{\pi {d^2}}} \times 100$
$
\Rightarrow \dfrac{{\pi {d^2} - \dfrac{{9\pi {d^2}}}{{16}}}}{{\pi {d^2}}} \times 100 \\
\Rightarrow \dfrac{{1 - \dfrac{9}{{16}}}}{1} \times 100 \\
\Rightarrow \dfrac{7}{{16}} \times 100 \\
\Rightarrow 43.75\% \\
$
So, the correct option is (a).
Note-Whenever we face such types of problems we use some important points. Like we know decrement in diameter so we use diameter in the formula of curved surface area of sphere rather than radius then after finding new curved surface area we can easily get percentage decreases in curved surface area.
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