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Question
The radius of a sphere increases by 25%. Find the percentage increase in its surface area
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Solution
Let the radius of the be r
Surface area of the sphere = 4πr2 sq.units ...(1)
If the radius is increased by 25%
New radius = `25/100 xx "r" + "r"`
= `"r"/4 + "r"`
= `("r" + 4"r")/4`
= `(5"r")/4`
Surface area of the sphere
= `4pi((5"r")/4)^2"sq.units"`
= `4 xx pi xx (25"r"^2)/16`
= `(25pi"r"^2)/4"sq.units"`
Difference in surface area
= `(25pi"r"^2)/4 - 4pi"r"^2`
= `pi"r"^2(25/4 - 4)`
= `pi"r"^2((25 - 16)/4)`
= `pi"r"^2(9/4)`
= `(9pi"r"^2)/4`
Percentage of increase in surface area
= `"Difference in surface area"/"Old surface area" xx 100`
= `((9pi"r"^2)/4)/(4pi"r"^2) xx 100`
= `(9pi"r"^2)/(4 xx 4pi"r"^2) xx 100`
= `9/16 xx 100%` = 56.25 %
Percentage of increase in surface area = 56.25 %
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