Question

The radius of a sphere increases by 25%. Find the percentage increase in its surface area

Answer 1

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 %