Problem $6$: Plotting on semi$-$log and log$-$log paper

Mathematical models frequently arise in the form of physical or empirical laws used

in science and engineering. For the two cases shown below, use the mathematical

models given to:

a$)$ Create a table of the independent $(x-$axis$)$ and dependent $(y-$axis$)$ values for

the ranges indicated. Include as a table in your solution $($with table number

and caption$).$

b$)$ Plot the data in Excel and include the plot as a figure in your solution $($with

figure number and caption$).$ Select the graph paper type that results in a

straight$-$line on the plot.

Case $1$: Ohms law is used with electrical circuits to determine the power $P($in W $)$ for

a given current I $($in amps$).$ The law has the form:

$P=R{I}^2$

where $R$ is the resistance $($in Ohms $).$ For an electrical circuit with a resistance $R$ of

$68$ Ohms, use the model to compute and plot $P$ for currents I of $0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}5,1\mathrm{.}0,2\mathrm{.}0,$

$5\mathrm{.}0$ and $10\mathrm{.}0$ amps $.$

Case $2$: Microorganisms in a culture dish grow exponentially until an essential

nutrient is exhausted. An empirical model for the growth predicts the mass of

microorganisms $M$ at time $t$ has the form:

$M=M_0{e}^{kt}$

where $M_0$ is the initial mass, and $k$ is the growth constant. Assuming that $M_0=0\mathrm{.}1g,$

and a growth constant $k=0\mathrm{.}3$ day ${?}^{-1},$ use the model to compute $M($in g$)$ at time steps

of $4$ days up to $28$ day $(0,4,8,$dots,$24,28$ days