Answer:
a) [tex]V = 33510.322\,in^{3}[/tex], b) [tex]A_{s} = 5026.548\,in^{2}[/tex], c) [tex]\% V = 0.450\,\%[/tex], [tex]\%A_{s} = 0.300\,\%[/tex].
Step-by-step explanation:
The volume and the surface area of the sphere are, respectively:
[tex]V = \frac{4}{3}\pi \cdot r^{3}[/tex]
[tex]A_{s} = 4\pi \cdot r^{2}[/tex]
a) The volume of the sphere is:
[tex]V = \frac{4}{3}\pi \cdot (20\,in)^{3}[/tex]
[tex]V = 33510.322\,in^{3}[/tex]
b) The surface area of the sphere is:
[tex]A_{s} = 4\pi \cdot (20\,in)^{2}[/tex]
[tex]A_{s} = 5026.548\,in^{2}[/tex]
c) The total differentials for volume and surface area of the sphere are, respectively:
[tex]\Delta V = 4\pi\cdot r^{2}\,\Delta r[/tex]
[tex]\Delta V = 4\pi \cdot (20\,in)^{2}\cdot (0.03\,in)[/tex]
[tex]\Delta V = 150.796\,in^{3}[/tex]
[tex]\Delta A_{s} = 8\pi\cdot r \,\Delta r[/tex]
[tex]\Delta A_{s} = 8\pi \cdot (20\,in)\cdot (0.03\,in)[/tex]
[tex]\Delta A_{s} = 15.080\,in^{2}[/tex]
Relative errors are presented hereafter:
[tex]\%V = \frac{\Delta V}{V}\times 100\%[/tex]
[tex]\%V = \frac{150.796 \,in^{3}}{33510.322\,in^{3}}\times 100\,\%[/tex]
[tex]\% V = 0.450\,\%[/tex]
[tex]\% A_{s} = \frac{\Delta A_{s}}{A_{s}}\times 100\,\%[/tex]
[tex]\% A_{s} = \frac{15.080\,in^{2}}{5026.548\,in^{2}}\times 100\,\%[/tex]
[tex]\%A_{s} = 0.300\,\%[/tex]
