Answer :

Given: a hollow spherical shell with internal radii 3cm and external radii 3.0005 cm

To find: the approximate volume of the metal in the hollow spherical shell

Explanation: Let the internal and external radii of the hollow spherical shell be r and R, respectively.

So it is given,

R = 3.0005 and r = 3

And let the volume of the hollow spherical shell be V.

Then we know,

Now substituting the values of R and r, we get

Now using the differentiation to get the approximate value of (3.0005)3.

But the integer nearest to 3.0005 is 3,

So 3.0005 = 3+0.0005

So let a = 3 and h = 0.0005

Hence, (3.0005)3 = (3+0.0005)3

Let the function becomes,

f(x) = x3………(ii)

Now applying first derivative, we get

f’(x) = 2x2……….(iii)

Now let f(a+h) = (3.0005)3

Now we know,

f(a+h) = f(a)+hf’(a)

Now substituting the function from (ii) and (iii), we get

f(a+h) = a3+h(3a2)

Substituting the values of a and h, we get

f(3+0.0005) = 33+(0.0005) (3(32))

f(3.0005) = 27+(0.0005)(3(9))

(3.0005)3 = 27+(0.0005)(27)

(3.0005)3 = 27+0.0135

(3.0005)3 = 27.0135

Hence the approximate value of (3.0005)3 = 27.0135.

Now substituting this in equation (i), we get

V = 4π(0.0045)

V = 0.018π cm3

Hence the approximate volume of the metal in the hollow spherical shell is 0.018π cm3.

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