Answer:
(a) The maximum error in the calculated surface area is 168 square centimeters, (b) The relative error is 7.48 per cent.
Step-by-step explanation:
(a) The surface area of a sphere ([tex]A_{s}[/tex]), measured in square centimeters, is represented by the following formula:
[tex]A_{s} = 4\pi\cdot r^{2}[/tex]
Where [tex]r[/tex] is the radius of the sphere, measured in centimeters.
The differential is obtained by this:
[tex]\Delta s = \frac{\partial A_{s}}{\partial r}\cdot \Delta r[/tex]
Where [tex]\frac{\partial A_{r}}{\partial r}[/tex] is the partial derivative of the surface area of the sphere in terms of the radius.
[tex]\frac{\partial A_{s}}{\partial r} = 8\pi \cdot r[/tex]
[tex]\Delta A_{s} = 8\pi \cdot r \cdot \Delta r[/tex]
The circumference ([tex]s[/tex]) of a sphere, measured in centimeters, is represented by this expression:
[tex]s =2\pi\cdot r[/tex]
Where [tex]r[/tex] is the radius of the sphere, measured in centimeters, and which is now cleared:
[tex]r = \frac{s}{2\pi}[/tex]
If [tex]s = 84\,cm[/tex], the radius of the sphere is:
[tex]r = \frac{84\,cm}{2\pi}[/tex]
[tex]r \approx 13.369\,cm[/tex]
Given that [tex]r \approx 13.369\,cm[/tex] and [tex]\Delta r = 0.5\,cm[/tex], the surface area of the sphere is:
[tex]\Delta A_{s} = 8\pi\cdot (13.369\,cm)\cdot (0.5\,cm)[/tex]
[tex]\Delta A_{s} \approx 168\,cm^{2}[/tex]
The maximum error in the calculated surface area is 168 square centimeters.
(b) The relative error, measured in percentage, is given by this equation:
[tex]\delta = \left(\frac{\Delta A_{s}}{A_{s}} \right)\times 100\,\%[/tex]
The surface area of the sphere is: ([tex]r \approx 13.369\,cm[/tex])
[tex]A_{s} = 4\pi\cdot (13.369\,cm)^{2}[/tex]
[tex]A_{s}\approx 2245.989\,cm^{2}[/tex]
If [tex]\Delta A_{s} \approx 168\,cm^{2}[/tex] and [tex]A_{s}\approx 2245.989\,cm^{2}[/tex], then:
[tex]\delta = \left(\frac{168\,cm^{2}}{2245.989\,cm^{2}} \right)\times 100\,\%[/tex]
[tex]\delta = 7.48\,\%[/tex]
The relative error is 7.48 per cent.
