In developing countries, it is common for people to cook food over an open fire inside their

partially$-$enclosed houses, which often leads to indoor air quality problems due to particulate matter $($PM$).$

A typical$-$sized house structure is $3$ m $5$ m $2\mathrm{.}5$ m and the PM emission rate from a cooking fire has been

reported to average $300$ g$/$s$.$ Make the following additional assumptions:

$$ PM decay rate coefficient is $1\mathrm{.}3310-4$ s$-1.$

$$ natural ventilation rate for a partially$-$enclosed house is $2\mathrm{.}25$ air changes per hour

$$ ambient outside air has a PM concentration of $5$ g$/$m$3$

a$)$ Before the cook fire is started, assume a house has an indoor PM concentration equal to the ambient

outside air. After the fire is started, what will be the PM concentration in the house after the fire has

been burning for $1$ hour? Report your answer in g$/$m$3.$

b$)$ If the fire continues to burn for a very long time $($e$.$g$.,$ time goes to infinity$),$ what will be the maximum

PM concentration in the house? Report your answer in g$/$m$3.$

c$)$ Assume the cook fire has burned for a very long time. Once the fire is extinguished and is no longer

emitting particulate matter, how long will it take for the PM concentration in the house to drop below

$50$ g$/$m$3($i$.$e$.,10$ the ambient outside air concentration$)?$ Report your answer to the nearest $0\mathrm{.}1$ h