Question

The drama club will give one performance every night except Sunday and Monday for twoweeks (ten nights). Club members believe that revenue from the first night's productionwill be approximately $3500. For each night after that, they think the revenue will beB70% of the previous night's revenue. Use this information to estimate projected revenuefor each of the first five nights of the production. Then write a function rule that modelsthis situation.

Answer 1

The revenue generated by the drama club play on the first night is given as:

$\text{\textcolor{#FF7968}{Day}}\textcolor{#FF7968}{1\colon}\text{ \ 3,500}$

The production teams projects the revenue earned each successive day to be 70% of the previous day. Therefore, we expect a 30% cut in revenue generated in each successive day for next five days.

Using the above information we can forecast the revenue to be generated in next 5 days as follows:

$\begin{gathered} \text{\textcolor{#FF7968}{Day 2:}}\text{ ( \ 3 , 500 }\cdot\text{ 0.7 ) = \ 2,450} \\ \text{\textcolor{#FF7968}{Day 3:}}\text{ (\ 2,450 }\cdot\text{ 0.7 ) = \ 1,715} \\ \text{\textcolor{#FF7968}{Day 4:}}\text{ (\ 1,715 }\cdot\text{ 0.7 ) = \ 1,}200.5 \\ \text{\textcolor{#FF7968}{Day 5:}}\text{ (\ 1,200.5 }\cdot\text{ 0.7 ) = \ }840.35 \\ \text{\textcolor{#FF7968}{Day 6:}}\text{ ( \ 840.35 }\cdot\text{ 0.7 ) = \ 588.245} \end{gathered}$

We can write the revenue for 6 days in a sequential form as follows:

$3500,2450,1715,1200.5,840.35,588.245,\ldots\text{ }$

We see that the above sequence follows a geometric progression. Where the parameters of geometric progression are as such:

$\begin{gathered} a\text{ = first term} \\ r\text{ = common ratio} \end{gathered}$

Where for this sequence the constant parameters are:

$a\text{ = \ 3500 , r = 0.7}$

The revenue earned ( Rn ) at the nth day till 5 days can be modeled by using geomtric progression nth terms formula as such:

$R_n\text{ = a}\cdot r^{n-1}$

By plugging in the respective constants ( a and r ) we can get the function rule that models the given situation as such:

$\textcolor{#FF7968}{R_n}\text{\textcolor{#FF7968}{ = \ 3500}}\textcolor{#FF7968}{\cdot(0.7)^{n\text{ - 1}}}$

Where, n = The nth consecutive day of the working week!