Chapter 3.4, Problem 29E

a.

To determine

Find the estimate of surface area of bacterium in terms of relative uncertainty.

Find the relative uncertainty in surface area of bacterium.

Answer to Problem 29E

The estimate of surface area of bacterium in terms of relative uncertainty is $\underset{\_}{S=17.59\mu \text{m}\pm 18\%}$.

The relative uncertainty in surface area of bacterium is $\underset{\_}{{\sigma }_{\ln \text{S}}=18\%}$.

Explanation of Solution

Given info:

The radius of a cylinder is measured to be $r=0.8\pm 0.1\mu \text{m}$ and height capped on each end of a hemisphere is measured to be $h=1.9\pm 0.1\mu \text{m}$.

Calculation:

The form of the measurements of a process is,

$\text{Measured}\text{value}\left(\mu \right)\pm \text{Standard deviation}\left(\sigma \right)$.

Here, for a random sample the measured value will be sample mean and the population standard deviation will be sample standard deviation.

The form of the measurements of radius of a cylinder is $r=0.8\pm 0.1\mu \text{m}$.

Here, the measured value of radius of a cylinder is $r=0.8\mu \text{m}$ and the uncertainty in the radius of a cylinder is ${\sigma }_r=0.1\mu \text{m}$.

The form of the measurements of height capped on each end of a hemisphere is $h=1.9\pm 0.1\mu \text{m}$.

Here, the measured value of height is $h=1.9\mu \text{m}$ and the uncertainty in the height is ${\sigma }_h=0.1\mu \text{m}$.

Measured value of surface area of bacterium:

The computational formula for surface area of the area is $S=2\pi r\left(h+2r\right)$.

Here, $h=1.9\mu \text{m}$ and $r=0.8\mu \text{m}$.

The measured value of surface area of bacterium is obtained as follows:

$\begin{array}{c}S=2\pi r\left(h+2r\right)\\ =2\times 3.14159\times 0.8\left(1.9+2\times 0.8\right)\\ =17.59\end{array}$

Thus, the measured value of surface area of bacterium is $\underset{\_}{S=17.59\mu \text{m}}$.

Uncertainty:

The uncertainty of a process is determined by the standard deviation of the measurements. In other words it can be said that, measure of variability of a process is known as uncertainty of the process.

Therefore, it can be said that uncertainty is simply $\left(\sigma \right)$.

Standard deviation:

The standard deviation is based on how much each observation deviates from a central point represented by the mean. In general, the greater the distances between the individual observations and the mean, the greater the variability of the data set.

The general formula for standard deviation is,

$s=\sqrt{\frac{{\displaystyle \sum x_i{}^2-\frac{{\left({\displaystyle \sum x_i}\right)}^2}{n}}}{n-1}}$.

From the properties of uncertainties for functions of one measurement it is known that,

• If $X_1,X_2,\mathrm{...},X_n$ are independent measurements with uncertainties ${\sigma }_{X_1},{\sigma }_{X_2},\mathrm{...},{\sigma }_{X_n}$ and if $U=U\left(X_1,X_2,\mathrm{..},X_n\right)$ is a function of $X_1,X_2,\mathrm{...},X_n$ then the uncertainty in the variable U is ${\sigma }_U=\sqrt{{\left(\frac{\partial U}{\partial X_1}\right)}^2{\sigma }_{X_1}^2+{\left(\frac{\partial U}{\partial X_2}\right)}^2{\sigma }_{X_2}^2+\mathrm{....}+{\left(\frac{\partial U}{\partial X_n}\right)}^2{\sigma }_{X_n}^2}$.

Here, radius and height are not constants. The surface area of bacterium is a function of radius and height.

Relative uncertainty:

Relative uncertainty in U is the uncertainty as a fraction of the true value (mean of the measurement ${\mu }_U$). Relative uncertainty is also called as coefficient of variation. Relative uncertainty is expressed in percentage without units.

The general formula to obtain relative uncertainty is,

${\sigma }_{\ln U}=\frac{{\sigma }_U}{{\mu }_U}=\frac{{\sigma }_U}{U}$.

Relative uncertainty in surface area of bacterium:

The relative uncertainty in surface area of bacterium is,

$\begin{array}{c}{\sigma }_{\ln \left(S=2\pi r\left(h+2r\right)\right)}=\sqrt{{\left(\frac{\partial \ln \left(\text{S}\right)}{\partial r}\right)}^2{\sigma }_r^2+{\left(\frac{\partial \ln \left(\text{S}\right)}{\partial h}\right)}^2{\sigma }_h^2}\\ =\sqrt{{\left(\frac{\partial \ln \left(2\pi r\left(h+2r\right)\right)}{\partial r}\right)}^2{\sigma }_r^2+{\left(\frac{\partial \ln \left(2\pi r\left(h+2r\right)\right)}{\partial h}\right)}^2{\sigma }_h^2}\\ =\sqrt{{\left(\frac{\partial \left(\ln \left(2\pi r\right)+\ln \left(h+2r\right)\right)}{\partial r}\right)}^2{\sigma }_r^2+{\left(\frac{\partial \left(\ln \left(2\pi r\right)+\ln \left(h+2r\right)\right)}{\partial h}\right)}^2{\sigma }_h^2}\\ =\sqrt{{\left(\frac{\partial \left(\ln \left(2\pi \right)+\ln \left(r\right)+\ln \left(h+2r\right)\right)}{\partial r}\right)}^2{\sigma }_r^2+{\left(\frac{\partial \left(\ln \left(2\pi \right)+\ln \left(r\right)+\ln \left(h+2r\right)\right)}{\partial h}\right)}^2{\sigma }_h^2}\end{array}$ $\begin{array}{c}=\sqrt{{\left(\frac{1}{r}+\frac{2}{h+2r}\right)}^2{\sigma }_r^2+{\left(\frac{1}{h+2r}\right)}^2{\sigma }_h^2}\\ =\sqrt{{\left(1.82143\right)}^2\times {\left(0.1\right)}^2+{\left(0.285714\right)}^2\times {\left(0.1\right)}^2}\\ =0.18\end{array}$

Thus, the relative uncertainty in surface area of bacterium is $\underset{\_}{{\sigma }_{\ln \text{S}}=18\%}$.

Estimate of surface area of bacterium in terms of relative uncertainty:

The estimate of the measurement of a process in terms of relative uncertainty is,

$\text{Measured}\text{value}\left(U\right)\pm \text{Standard deviation}\left({\sigma }_{\ln U}\right)$.

The estimate of surface area of bacterium in terms of relative uncertainty is,

$\begin{array}{c}S=\text{Measured value of S}\pm {\sigma }_{\ln S}\\ =\underset{\_}{17.59\mu \text{m}\pm 18\%}\end{array}$

Thus, the estimate of surface area of bacterium in terms of relative uncertainty is $\underset{\_}{S=17.59\mu \text{m}\pm 18\%}$.

b.

To determine

Find the estimate of volume of bacterium in terms of relative uncertainty.

Find the relative uncertainty in volume of bacterium.

Answer to Problem 29E

The estimate of volume of bacterium in terms of relative uncertainty is $\underset{\_}{\text{V}=5.965\mu {\text{m}}^3\pm 30\%}$.

The relative uncertainty in volume of bacterium is $\underset{\_}{{\sigma }_{\ln \text{V}}=30\%}$.

Explanation of Solution

Calculation:

From part (a), $r=0.8\mu \text{m}$, ${\sigma }_r=0.1\mu \text{m}$ and $h=1.9\mu \text{m}$, ${\sigma }_h=0.1\mu \text{m}$.

Measured value of volume of bacterium:

The computational formula for volume of bacterium is $V=\pi r^2\left(h+\frac{4r}{3}\right)$.

Here, $h=1.9\mu \text{m}$ and $r=0.8\mu \text{m}$.

The measured value of volume of bacterium is obtained as follows:

$\begin{array}{c}V=\pi r^2\left(h+\frac{4r}{3}\right)\\ =3.14159\times {0.8}^2\left(1.9+\frac{4\times 0.8}{3}\right)\\ =5.965\end{array}$

Thus, the measured value of volume of bacterium is $\underset{\_}{V=5.965\mu {\text{m}}^3}$.

Relative uncertainty in volume of bacterium:

The relative uncertainty in volume of bacterium is,

$\begin{array}{c}{\sigma }_{\ln \left(V=\pi r^2\left(h+\frac{4r}{3}\right)\right)}=\sqrt{{\left(\frac{\partial \ln \left(V\right)}{\partial r}\right)}^2{\sigma }_r^2+{\left(\frac{\partial \ln \left(V\right)}{\partial h}\right)}^2{\sigma }_h^2}\\ =\sqrt{{\left(\frac{\partial \ln \left(\pi r^2\left(h+\frac{4r}{3}\right)\right)}{\partial r}\right)}^2{\sigma }_r^2+{\left(\frac{\partial \ln \left(\pi r^2\left(h+\frac{4r}{3}\right)\right)}{\partial h}\right)}^2{\sigma }_h^2}\\ =\sqrt{{\left(\frac{\partial \left(\ln \left(\pi r^2\right)+\ln \left(h+\frac{4r}{3}\right)\right)}{\partial r}\right)}^2{\sigma }_r^2+{\left(\frac{\partial \left(\ln \left(\pi r^2\right)+\ln \left(h+\frac{4r}{3}\right)\right)}{\partial h}\right)}^2{\sigma }_h^2}\\ =\sqrt{{\left(\frac{\partial \left(\ln \left(\pi \right)+2\ln \left(r\right)+\ln \left(h+\frac{4r}{3}\right)\right)}{\partial r}\right)}^2{\sigma }_r^2+{\left(\frac{\partial \left(\ln \left(\pi \right)+2\ln \left(r\right)+\ln \left(h+\frac{4r}{3}\right)\right)}{\partial h}\right)}^2{\sigma }_h^2}\end{array}$ $\begin{array}{c}=\sqrt{{\left(\frac{2}{r}+\frac{4}{\left(3h+4r\right)}\right)}^2{\sigma }_r^2+{\left(\frac{2}{r}+\frac{4}{\left(3h+4r\right)}\right)}^2{\sigma }_h^2}\\ =\sqrt{{\left(2.94944\right)}^2\times {\left(0.1\right)}^2+{\left(0.337079\right)}^2\times {\left(0.1\right)}^2}\\ =0.3\end{array}$

Thus, the relative uncertainty in volume of bacterium is $\underset{\_}{{\sigma }_{\ln \text{V}}=30\%}$.

Estimate of volume of bacterium in terms of relative uncertainty:

The estimate of the measurement of a process in terms of relative uncertainty is,

$\text{Measured}\text{value}\left(U\right)\pm \text{Standard deviation}\left({\sigma }_{\ln U}\right)$.

The estimate of volume of bacterium in terms of relative uncertainty is,

$\begin{array}{c}\text{V}=\text{Measured value of V}\pm {\sigma }_{\ln V}\\ =5.965\mu {\text{m}}^3\pm 30\%\end{array}$

Thus, the estimate of volume of bacterium in terms of relative uncertainty is $\underset{\_}{\text{V}=5.965\mu {\text{m}}^3\pm 30\%}$.

c.

To determine

Find the estimate of rate at which a chemical is absorbed into the bacterium in terms of relative uncertainty.

Find the relative uncertainty in the rate at which a chemical is absorbed into the bacterium.

Answer to Problem 29E

The estimate of rate at which a chemical is absorbed into the bacterium in terms of relative uncertainty is $\underset{\_}{R=2.95c\frac{1}{{\text{m}}^2}\pm 11\%}$.

The relative uncertainty in the rate at which a chemical is absorbed into the bacterium is $\underset{\_}{{\sigma }_{\ln R}=11\%}$.

Explanation of Solution

Calculation:

The computation formula for the volume of the bacterium is $V=\pi r^2\left(h+\frac{4r}{3}\right)$ and the computational formula for surface area of the area is $S=2\pi r\left(h+2r\right)$.

Measured value of rate at which a chemical is absorbed into the bacterium:

The formula for rate at which a chemical is absorbed into the bacterium is $R=c\times \left(\frac{S}{V}\right)$.

Here, $V=\pi r^2\left(h+\frac{4r}{3}\right)$ and $S=2\pi r\left(h+2r\right)$.

The computation formula for R is,

$\begin{array}{c}R=c\times \left(\frac{S}{V}\right)\\ =c\times \left(\frac{2\pi r\left(h+2r\right)}{\pi r^2\left(h+\frac{4r}{3}\right)}\right)\\ =c\times \left(\frac{\left(2h+4r\right)}{\left(hr+\frac{4r^2}{3}\right)}\right)\end{array}$

Thus, the computation formula for R is $R=c\times \left(\frac{\left(2h+4r\right)}{\left(hr+\frac{4r^2}{3}\right)}\right)$

From part (a), $r=0.8\mu \text{m}$, ${\sigma }_r=0.1\mu \text{m}$ and $h=1.9\mu \text{m}$, ${\sigma }_h=0.1\mu \text{m}$.

The measured value of rate at which a chemical is absorbed into the bacterium is obtained as follows:

$\begin{array}{c}R=c\times \left(\frac{\left(2h+4r\right)}{\left(hr+\frac{4r^2}{3}\right)}\right)\\ =c\times \left(\frac{\left(\left(2\times 1.9\right)+\left(4\times 0.8\right)\right)}{\left(1.9\times 0.8+\frac{4\times {0.8}^2}{3}\right)}\right)\\ =c\times \left(\frac{7}{2.3733}\right)\\ =2.95c\end{array}$

Thus, the measured value of rate at which a chemical is absorbed into the bacterium is $\underset{\_}{R=2.95c}$.

Here, radius and height are not constants. The rate at which a chemical is absorbed into the bacterium is a function of radius and height.

Relative uncertainty in the rate at which a chemical is absorbed into the bacterium:

The relative uncertainty in the rate at which a chemical is absorbed into the bacterium is,

${\sigma }_{\ln \left(R=c\times \left(\frac{\left(2h+4r\right)}{\left(hr+\frac{4r^2}{3}\right)}\right)\right)}=\sqrt{{\left(\frac{\partial \ln \left(R\right)}{\partial r}\right)}^2{\sigma }_r^2+{\left(\frac{\partial \ln \left(R\right)}{\partial h}\right)}^2{\sigma }_h^2}$

The value of $\frac{\partial \ln \left(R\right)}{\partial r}$:

The value of $\frac{\partial \ln \left(R\right)}{\partial r}$ is obtained as follows:

$\begin{array}{c}\frac{\partial \ln \left(R\right)}{\partial r}=\frac{\partial \ln \left(c\times \left(\frac{\left(2h+4r\right)}{\left(hr+\frac{4r^2}{3}\right)}\right)\right)}{\partial r}\\ =\frac{\partial \left(\ln \left(c\right)+\ln \left(\frac{\left(2h+4r\right)}{\left(hr+\frac{4r^2}{3}\right)}\right)\right)}{\partial r}\\ =\frac{\partial \left(\ln \left(c\right)+\ln \left(2h+4r\right)-\ln \left(hr+\frac{4r^2}{3}\right)\right)}{\partial r}\\ =\frac{4}{2h+4r}-\frac{1}{\left(hr+\frac{4r^2}{3}\right)}\left(h+\frac{8r}{3}\right)\end{array}$

           $\begin{array}{c}=\frac{4}{2h+4r}-\frac{3h+8r}{3hr+8r^2}\\ =\frac{1}{0.8}-\frac{3\times 1.9+8\times 0.8}{3\times 1.9\times 0.8+8\times {0.8}^2}\\ =-1.12801\end{array}$

The value of $\frac{\partial \ln \left(R\right)}{\partial h}$:

The value of $\frac{\partial \ln \left(R\right)}{\partial h}$ is obtained as follows:

$\begin{array}{c}\frac{\partial \ln \left(R\right)}{\partial h}=\frac{\partial \ln \left(c\times \left(\frac{\left(2h+4r\right)}{\left(hr+\frac{4r^2}{3}\right)}\right)\right)}{\partial h}\\ =\frac{\partial \left(\ln \left(c\right)+\ln \left(\frac{\left(2h+4r\right)}{\left(hr+\frac{4r^2}{3}\right)}\right)\right)}{\partial h}\\ =\frac{\partial \left(\ln \left(c\right)+\ln \left(2h+4r\right)-\ln \left(hr+\frac{4r^2}{3}\right)\right)}{\partial h}\\ =\frac{2}{2h+4r}-\frac{1}{\left(hr+\frac{4r^2}{3}\right)}\left(r\right)\end{array}$

$\begin{array}{c}=\frac{2}{2h+4r}-\frac{3r}{3hr+4r^2}\\ =\frac{2}{2\times 1.9+4\times 0.8}-\frac{3\times 1.9}{3\times 1.9\times 0.8+4\times {0.8}^2}\\ =-0.0513644\end{array}$

The relative uncertainty in the rate at which a chemical is absorbed into the bacterium is,

$\begin{array}{c}{\sigma }_{\ln \left(R=c\times \left(\frac{\left(2h+4r\right)}{\left(hr+\frac{4r^2}{3}\right)}\right)\right)}=\sqrt{{\left(\frac{\partial \ln \left(R\right)}{\partial r}\right)}^2{\sigma }_r^2+{\left(\frac{\partial \ln \left(R\right)}{\partial h}\right)}^2{\sigma }_h^2}\\ =\sqrt{{\left(-1.12801\right)}^2\times {\sigma }_r^2+{\left(-0.0513644\right)}^2\times {\sigma }_h^2}\\ =\sqrt{{\left(-1.12801\right)}^2\times {\left(0.1\right)}^2+{\left(-0.0513644\right)}^2\times {\left(0.1\right)}^2}\\ =0.11\end{array}$

Thus, the relative uncertainty in the rate at which a chemical is absorbed into the bacterium is $\underset{\_}{{\sigma }_{\ln R}=11\%}$.

Estimate of the rate at which a chemical is absorbed into the bacterium in terms of relative uncertainty:

The estimate of the measurement of a process in terms of relative uncertainty is,

$\text{Measured}\text{value}\left(U\right)\pm \text{Standard deviation}\left({\sigma }_{\ln U}\right)$.

The estimate of rate at which a chemical is absorbed into the bacterium in terms of relative uncertainty is,

$\begin{array}{c}R=\text{Measured value of }R\pm {\sigma }_{\ln R}\\ =2.95c\frac{1}{{\text{m}}^2}\pm 11\%\end{array}$

Thus, the estimate of rate at which a chemical is absorbed into the bacterium in terms of relative uncertainty is $\underset{\_}{R=2.95c\frac{1}{{\text{m}}^2}\pm 11\%}$.

d.

To determine

Check whether the relative uncertainty in R depend on c..

Answer to Problem 29E

No, the relative uncertainty in R is independent on c.

Explanation of Solution

Calculation:

The formula for rate at which a chemical is absorbed into the bacterium is $R=c\times \left(\frac{S}{V}\right)$.

Here, $V=\pi r^2\left(h+\frac{4r}{3}\right)$ and $S=2\pi r\left(h+2r\right)$.

Here, radius and height are not constants. The rate at which a chemical is absorbed into the bacterium is a function of radius and height and c is a constant.

Therefore, the uncertainty in c is “0”.

The relative uncertainty in the rate at which a chemical is absorbed into the bacterium is ${\sigma }_{\ln R}=11\%$.

Therefore, the relative uncertainty in R is independent on c.