smc-1185-30-Area

Chemicals are added to a container so that a particular crystal will grow in the shape of a cube. The side length of the crystal, `x` millimetres, `t` days after the chemicals were added to the container, is given by `x = arctan(t)`.

Find the rate at which the surface area, `A` square millimetres, of the crystal is growing one day after the chemicals were added. Give your answer in square millimetres per day. (4 marks)

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`(3 pi)/2`

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`x`	`=tan^(-1) x`
`(dx)/(dt)`	`= 1/(1 + t^2)`

`text(Surface Area)\ \ (A) = 6x^2`

`(dA)/(dx)`	`= 12x`
`(dA)/(dt)`	`= (dA)/(dx) ⋅ (dx)/(dt)`
	`= (12 x)/(1 + t^2)`

`text(When)\ \ t=1,`

`x= tan^(-1) (1) = pi/4`

`:. (dA)/(dt) \|_(x=pi/4,t=1)`	`=(12 xx pi/4)/(1+1^2)`
	`=(3pi)/2\ text(mm²/day)`