The water level in a retention pond started a 5 ft (60 in.) and decreased at a rate of 2 in./day during a 14-day drought. A tropical depression moved through at the beginning of the 15 day and proceed rain at an average rate of 2.4 in./day for 5 days.(a) Write a piecewise - defined function to model the water level L(x) (in inches) as a function of the number of days x since the beginning of the drought.(b) Graph the function.

Since the initial level of the pond at the start of our model is 60 inches and the rate of decrease is 2 inches per day the model can be replaced by a straight line

[tex]y=mt+c[/tex]

where

'y' is the water level in the pool at any time 't'

'C' is the water level at the start given as 60 inches

'm' is the rate of decrease of the water level given as -2 inches per day

't' is the time in days from the start of the model thus applying the given values we get

[tex]y(t)=\left\{\begin{matrix}60-2t\end{matrix}\right.(t\leq 14)[/tex]

The water level at the end of the 14 days is

[tex]y(14)=60-2\times 14=32[/tex]

Now since after 14 days the water level begins to rise at a rate 2.4 inches per day from a balance of 32 inches the water level is similarly modelled as above

[tex]y'(t)=\left\{\begin{matrix}32+2.4t\end{matrix}\right.(t\geq 14)[/tex]

Thus the combined function can be represented as

[tex]y(t)=\left\{\begin{matrix}(60-2t)(0< t\leq 14)\\\\32+2.4t(t\geq 14) & \\ & \end{matrix}[/tex]

The graph is shown in the attached figure below