A composition sensor is used to continuously monitor the contaminant level in a liquid stream.

The dynamic behavior of the sensor can be described by a first$-$order transfer function with a

time constant of $12$ seconds.

$\frac{C_m^{\text{'}}(s)}{C^{\text{'}}(s)}=\frac{1}{12s+1}$

where $C^{\text{'}}$ is the actual contaminant concentration and $C_m^{\text{'}}$ is the measured value. Both are

expressed as deviation variables $($e$.$g$.C^{\text{'}}=C\frac{-}{b}ar(C)).$ Both $C$ and $C_m$ have concentration values of $8$

ppm$,$ nominally.

An alarm sounds if the measured value exceeds the environmental value of $13ppm.$ Suppose

that the contaminant concentration $C$ gradually increases according to the expression:

$C(t)=8+0\mathrm{.}4t$

where $t$ is expressed in seconds.

a$.$ How long $($in seconds$)$ will it take for the actual concentration of the contaminant in the

liquid stream to reach $13ppm?$

b$.$ How long $($in seconds$)$ will it take for the measured concentration of the contaminant in the

liquid stream to reach $13$ ppm$?$ Hint: Use Excel Solver.

c$.$ How long $($in seconds$)$ will it take after the actual concentration of the contaminant in the

liquid stream has reached $13ppm$ for the alarm to sound?