Respuesta :
Answer:
(a) Approximate the percent error in computing the area of the circle: 4.5%
(b) Estimate the maximum allowable percent error in measuring the circumference if the error in computing the area cannot exceed 3%: 0.6 cm
Step-by-step explanation:
(a)
First we need to calculate the radius from the circumference:
[tex]c=2\pi r\\r=\frac{c}{2\pi } \\c=8.9 cm[/tex]
I leave only one decimal as we need to keep significative figures
Now we proceed to calculate the error for the radius:
[tex]\Delta r=\frac{dt}{dc} \Delta c\\\\\frac{dt}{dc} = \frac{1}{2 \pi } \\\\\Delta r=\frac{1}{2 \pi } (1.2)\\\\\Delta r= 0.2 cm[/tex]
[tex]r = 8.9 \pm 0.2 cm[/tex]
Again only one decimal because the significative figures
Now that we have the radius, we can calculate the area and the error:
[tex]A=\pi r^{2}\\A=249 cm^{2}[/tex]
Then we calculate the error:
[tex]\Delta A= (\frac{dA}{dr} ) \Delta r\\\\\Delta A= 2\pi r \Delta r\\\\\Delta A= 11.2 cm^{2}[/tex]
[tex]A=249 \pm 11.2 cm^{2}[/tex]
Now we proceed to calculate the percent error:
[tex]\%e =\frac{\Delta A}{A} *100\\\\\%e =\frac{11.2}{249} *100\\\\\%e =4.5\%[/tex]
(b)
With the previous values and equations, now we set our error in 3%, so we just go back changing the values:
[tex]\%e =\frac{\Delta A}{A} *100\\\\3\%=\frac{\Delta A}{249} *100\\\\\Delta A =7.5 cm^{2}[/tex]
Now we calculate the error for the radius:
[tex]\Delta r= \frac{\Delta A}{2 \pi r}\\\\\Delta r= \frac{7.5}{2 \pi 8.9}\\\\\Delta r= 0.1 cm[/tex]
Now we proceed with the error for the circumference:
[tex]\Delta c= \frac{\Delta r}{\frac{1}{2\pi }} = 2\pi \Delta r\\\\\Delta c= 2\pi 0.1\\\\\Delta c= 0.6 cm[/tex]
