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.

Explanation

Let at any time,`x` be the radius and `y` be the area of the plate,
then `y=pix^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,
`(Delta x)/x xx10=2`
when, `x=10`,
`(Delta x)/x xx10=2`
`=>(Delta x)/10 xx10=2`
`=>Delta x=2/10`
`=>dx=2/10`
now,
`y=pix^2`
`=>(dy)/(dx)=2pi x `
`=>((dy)/(dx))_(x=10)=20pi`
`therefore dy=(dy)/(dx)dx`
`=>dy=20 pixx 2/10`
`=4 pi`
so,`Delta y= 4 pi`
hence the approx. change in the area of the plate is `4pi`