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!