Question

If the circumference of a circle is increased by 40%,

then the area of the circle is increased by
A. 18%. B. 20%. C. 40%. D. 96%.

Answer 1

`r_(2)`Answer:

D) 96%

Explanation:

let the circumference of circle be C = 2π
`r_(1)` and
`r_(1)` =
`d_(1)`/2

Initial radius (radius before increment) =
`r_(1)`

initial diameter (diameter before increment) =
`d_(1)`

Increasing the circumference of a circle other factors are constant except the radius which can be used to find the diameter hence increasing the circumference of a circle is equivalent to increasing the diameter

area of the circle before increment
`A_(1)` = π
`r_(1)`²

`A_(1)` = π × (
`d_(1)`/2)²

`A_(1)` = π
`d_(1)`²/4

Increment of circumference results to increment of only the diameter

diameter after increment by 40%,
`d_(2)` = 1.4
`d_(1)`

1.4 represents increment by 40%

radius after increment
`r_(2)` =
`d_(2)`/2 = 1.4
`d_(1)`/2 = 0.7
`d_(1)`

area after increment,
`A_(2)` = π
`r^(2) _(2)`

`A_(2)` = π(0.7
`d_(1)`)²

`A_(2)` = π × 0.49
`d_(1)`²

Change in area =
`A_(2)` -
`A_(1)`

`A_(2)` -
`A_(1)` = π × 0.49
`d_(1)`² - π
`d_(1)`²/4

`A_(2)` -
`A_(1)` = π
`d_(1)`²(0.49 - ¼)

`A_(2)` -
`A_(1)` = π
`d_(1)`²(0.49 - 0.25)

`A_(2)` -
`A_(1)` = 0.24 π
`d_(1)`²

Percentage increment = (
`A_(2)` -
`A_(1)`/
`A_(1)`) × 100

= ((0.24 π
`d_(1)`²)/(π
`d_(1)`²/4)) × 100

= (0.24 × 4 ) × 100

= 0.96 × 100

= 96 %

Hope it was useful

Answer 2

96%

Explanation:

c=100=22/7D

D=7/22×100

get the radius and find the area

R=D/2

A=pi×r^2

repeat that assuming

C=140

get the area

find the %age

A2_A1=Area difference

A(diff)/A1×100=96.01%