Answer:
A) The maximum error in the calculated surface area: [tex]25cm^2[/tex]
Relative error: [tex]0.013[/tex]
B) The maximum error in the calculated volume: [tex]162cm^2[/tex]
Relative error: [tex]0.019[/tex]
Step-by-step explanation:
A) The formula for the surface area is:
[tex]A=4\pi r^2[/tex]
The measured value is the circumference which is equal to:
[tex]C=2\pi r[/tex]
then the radius is:
[tex]r=\frac{C}{2\pi}[/tex]
Substituting in the formula of the surface:
[tex]A=4\pi(\frac{C}{2\pi})^2\\A=4\pi(\frac{C^2}{4\pi^2})\\A=\frac{C^2}{\pi}[/tex]
Using the formula to calculate the error:
[tex]dy=f'(x)dx[/tex]
Where [tex]x[/tex] is the variable measured and [tex]y[/tex] is a function of [tex]x[/tex]([tex]y=f(x)[/tex]).
[tex]dA=f'(C)dC\\dA=\frac{2C^{(2-1)}}{\pi}dC\\dA=\frac{2C}{\pi}dC[/tex]
We have C=80cm and dC=0.5cm
[tex]dA=\frac{2C}{\pi}dC\\dA=\frac{2(80)}{\pi}(0.5)\\dA=\frac{160}{\pi}(0.5)\\dA=50.9296(0.5)\\dA=25.4648\approx25cm^2[/tex]
The relative error is the maximum error divide by the total area. The total area is: [tex]A=\frac{C^2}{\pi}=\frac{(80)^2}{\pi}=\frac{6400}{\pi}=2037.1833cm^2[/tex]
[tex]\frac{dA}{A}=\frac{25.4648}{2037.1833} =0.0125\approx0.013[/tex]
B) The formula for the volume is:
[tex]V=\frac{4}{3} \pi r^3[/tex]
Using [tex]r=\frac{C}{2\pi}[/tex]
[tex]V=\frac{4}{3} \pi r^3\\V=\frac{4}{3} \pi (\frac{C}{2\pi})^3\\V=\frac{4}{3} \pi (\frac{C^3}{8\pi^3})\\V=\frac{1}{3}(\frac{C^3}{2\pi^2})\\V=\frac{C^3}{6\pi^2}[/tex]
The maximum error is:
[tex]dV=\frac{3C^{3-1}}{6\pi^2}dC\\dV=\frac{C^{2}}{2\pi^2}dC\\dV=\frac{(80)^{2}}{2\pi^2}(0.5)\\dV=\frac{6400}{2\pi^2}(0.5)\\dV=\frac{6400}{2\pi^2}(0.5)\\dV=(324.2278)(0.5)\\dV=162.1139\approx162cm^2[/tex]
The calculated volume is:
[tex]V=\frac{C^3}{6\pi^2}\\V=\frac{(80)^3}{6\pi^2}\\V=\frac{512000}{6\pi^2}\\V=8646.0743[/tex]
The relative error is:
[tex]\frac{dV}{V}=\frac{162.1139}{8646.0743}=0.0188\approx0.019[/tex]
