To solve this problem we will apply the concepts related to the calculation of the surface, volume and error through the differentiation of the formulas given for the calculation of these values in a circle. Our values given at the beginning are
[tex]\phi = 76cm[/tex]
[tex]Error (dr) = 0.5cm[/tex]
The radius then would be
[tex]\phi = 2\pi r \\76cm = 2\pi r\\r = \frac{38}{\pi} cm[/tex]
And
[tex]\frac{d\phi}{dr} = 2\pi \\d\phi = 2\pi dr \\0.5 = 2\pi dr[/tex]
PART A ) For the Surface Area we have that,
[tex]A = 4\pi r^2 \\A = 4\pi (\frac{38}{\pi})^2\\A = \frac{5776}{\pi}[/tex]
Deriving we have that the change in the Area is equivalent to the maximum error, therefore
[tex]\frac{dA}{dr} = 4\pi (2r) \\dA = 4r (2\pi dr)[/tex]
Maximum error:
[tex]dA = 4(\frac{38}{\pi})(0.5)[/tex]
[tex]dA = \frac{76}{\pi}cm^2[/tex]
The relative error is that between the value of the Area and the maximum error, therefore:
[tex]\frac{dA}{A} = \frac{\frac{76}{\pi}}{\frac{5776}{\pi}}[/tex]
[tex]\frac{dA}{A} = 0.01315 = 1.31\%[/tex]
PART B) For the volume we repeat the same process but now with the formula for the calculation of the volume in a sphere, so
[tex]V = \frac{4}{3} \pi r^3[/tex]
[tex]V = \frac{4}{3} \pi (\frac{38}{\pi})^3[/tex]
[tex]V = \frac{219488}{3\pi^2}[/tex]
Therefore the Maximum Error would be,
[tex]\frac{dV}{dr} = \frac{4}{3} 3\pi r^2[/tex]
[tex]dV = 2r^2 (2\pi dr)[/tex]
[tex]dV = 4r^2 (\pi dr)[/tex]
Replacing the value for the radius
[tex]dV = 4(\frac{38}{\pi})^2(0.5)[/tex]
[tex]dV = \frac{2888}{\pi^2} cm^3[/tex]
And the relative Error
[tex]\frac{dV}{V} = \frac{ \frac{2888}{\pi^2}}{ \frac{219488}{3\pi^2} }[/tex]
[tex]\frac{dV}{V} = 0.03947[/tex]
[tex]\frac{dV}{V} = 3.947\%[/tex]
