A circular metal plate expands under heating so that its radius increases by 2%. Find the approximate increase in the area of the plate if the radius of the plate before heating is 10 cm.

Options

A. `2pi`
B. `3pi`
C. `pi`
D. `4pi`

Answer

D

Explanation

Let any time, x be the radius and y be the area of the plate. Then, `y=pi x^(2)`.
Let `Delta x` be the change in the radius and let `Delta y'` be the corresponding change in the area of the plate. Then,`(Deltax)/x xx100=2`
When `x=10`,
`(Deltax)/x xx100 = 2 rArr (Deltax)/10 xx 100 =2 rArr Deltax =2/10` `dx=2/10 " "[:' dx cong Delta x]...(i)`
Now, `y=y=pi x^(2) rArrdy/dx = 2pi " "x rArr(dy/dx)_(x=10)=20pi`
`:. " "dy=dy/dx dx`
`rArr " "dy=20pixx2/10= 4pi rArr Delta y=4pi " "[:' " "dy cong Delta " y]`
Hence, the approimate change in the area of the plate is `4pi`. SOME USEFUL RESULTS ON DIFFERENTIALS
(i) If(x) is a constant function, then its differential is zero.
(ii) If y = cu where c is a constant, then `dy= c " "du`
(iii) If `y=uy=pmv, then" "dy=du pm dv`
(iv) If `y=uv, then " "dy=u" "dv +v" "du`
(v) If `y=u/v, then" "dy=(v" "du-u" "dv)/v^(2)`
(vi) If `y=f(x), then" "dy = f'(x) dx`