We can use differentiation to tackle this kind of problem.

For a sphere,

V = Volume and A= Area,

`V = (4pir^3)/3 and A = 4pir^2`

now we can differentiate these two functions with respect to t, to find the rates.

Differentiating V,

`(dV)/(dt) = (4pi)/3*3r^2*(dr)/(dt)`

`(dV)/(dt) = 4pir^2(dr)/(dt)`

Differentiating...

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We can use differentiation to tackle this kind of problem.

For a sphere,

V = Volume and A= Area,

`V = (4pir^3)/3 and A = 4pir^2`

now we can differentiate these two functions with respect to t, to find the rates.

Differentiating V,

`(dV)/(dt) = (4pi)/3*3r^2*(dr)/(dt)`

`(dV)/(dt) = 4pir^2(dr)/(dt)`

Differentiating A,

`(dA)/(dt) = 4pi*2r*(dr)/(dt)`

`(dA)/(dt) = 8pir(dr)/(dt)`

When,

` r = 10^6 km, (dr)/(dt) = -500 kms^-1`

Therefore the rate of volume shrinkage,

`(dV)/(dt) = 4pi(10^6)^2*(-500) = -6.283*10^15`

So the rate of volume shrinkage is 6.283 x 10^15 km^3s^-1

The rate of area shrinkage,

`(dA)/(dt) = 8pi*10^6*(-500) = -1.257*10^10`

Therefore the rate of area shrinkage is 1.257 x 10^10 km^2s^-1