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Q.

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

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a

43.75%

b

21.88%

c

50%

d

25%

answer is A.

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Detailed Solution

Detailed solution can be understood by knowing the formula of csa and also the percentage calculations.
Concept- The formula for determining the curved surface area (CSA) of a sphere is
C.S.A = 4πr2.
C.S.A = The sum of areas of all the faces of a 3D object is called as Let us assume the diameter of the sphere is “d”.
So, the radius of the sphere in form of diameter is as given:
r =

\frac{d}{2}

The C.S.A. of the sphere = 4πr2 = 4π

{(\frac{d}{2})}^{2}

Therefore, the C.S.A of the sphere = πd2
Decrease percentage in diameter = 25%
⇒ 25% of d = d

\times

\frac{25}{100}

=

\frac{d}{4}

⇒

\frac{d}{4}

will be subtracted from total diameter (d).
So, the new diameter will be D = d -

\frac{d}{4}

=

\frac{3 d}{4}

Hence, the new curved surface area of the sphere = πD2
⇒ π

{(\frac{3 d}{4})}^{2}

⇒

\frac{9 π d^{2}}{16}

Percentage decrease in the C.S.A =

\frac{π d^{2} - π D^{2}}{π d^{2}}

\times

100
⇒

\frac{π d^{2} - π D^{2}}{π d^{2}}

\times

100
⇒

\frac{1 - \frac{9}{16}}{1}

\times

100
⇒

\frac{7}{16}

\times

100
⇒ 43.75%
Hence, the correct option is 1.

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