A clay bowl is in the shape of hollow hemisphere.
The external radius of the bowl is 8.2 cm. The internal radius of the bowl is 7.7cm. Both measurements are correct to the nearest 0.1cm.
the upper bound for the volume of clay is $${\mathtt{k}}{\mathtt{\,\times\,}}{{\mathtt{\pi}}}^{{\mathtt{3}}}$$ (cm).
Find the exact value of k.
Could you also state which topic this question is on, as I was told to find out.
+33661 The volume of a hemisphere is given by
$$volume=\frac{2}{3}\pi r^3$$
where r is the radius.
The desired volume of clay will be the difference between the outer and inner hemispherical volumes, so:
$$k\pi^3=\frac{2}{3}\pi8.2^3-\frac{2}{3}\pi7.7^3 = \frac{2}{3}\pi(8.2^3-7.7^3)$$
The exact value of k is therefore
$$k=\frac{2}{3}\frac{(8.2^3-7.7^3)}{\pi^2}$$
Numerically, the approximate value is:
$${\mathtt{k}} = {\frac{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right){\mathtt{\,\times\,}}\left({{\mathtt{8.2}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{{\mathtt{7.7}}}^{{\mathtt{3}}}\right)}{{{\mathtt{\pi}}}^{{\mathtt{2}}}}} \Rightarrow {\mathtt{k}} = {\mathtt{6.405\: \!862\: \!967\: \!147\: \!401\: \!7}}$$
or k ≈ 6.41 (it will have units of cm3)
I would say this topic is Geometry.
