Q. 89 - Precalculus Enhanced with Graphing Utilities

Question

The daily profits of a cigar company from selling x cigars are given by

$p\left(x\right)=-0.05x^2+100x-2000$

The government wishes to impose a tax on cigars (sometimes

called a $\sin$ tax) that gives the company the option of

either paying a flat tax of \($10000$ per day or a tax of $10$% on

profits. As chief financial officer (CFO) of the company, you

need to decide which tax is the better option for the company.

(a) On the same screen, graph $Y_1=p\left(x\right)-10000$ and $Y_2=\left(1-0.10\right)p\left(x\right)$.

(b) Based on the graph, which option would you select?

Why?

(c) Using the terminology learned in this section, describe

each graph in terms of the graph of $p\left(x\right)$.

(d) Suppose that the government offered the options of a

flat tax of \)$4800$ or a tax of 10% on profits. Which would

you select? Why

Step-by-Step Solution

Answer

(a) Graph of $Y_1=-0.05x^2+100x-12000 and Y_2=-0.045x^2+90x-1800$ is

(b) I would select paying tax of  $10$% of the profit.

(c) Graph of $Y_1$ is equivalent to the vertical downward shift of graph of$p\left(x\right)$ by$10000$ units

Graph of $Y_2$ is equivalent to the vertical compression of the graph of $p\left(x\right)$by the factor of $0.9$

(d) I would select paying tax of $10$% of the profit.

Step 1. Given information

Function for The daily profits of a cigar company from selling x cigars is

$p\left(x\right)=-0.05x^2+100x-2000$

Function for profit after paying flat tax of $$10000$ per day is

$Y_1=p\left(x\right)-10000$

Function for profit after paying tax of $10$% of profit is

$Y_2=\left(1-0.10\right)p\left(x\right)$

Step 2. Part (a)

profit after paying flat tax of $$1000$ per day

$$\begin{gathered} Y_1=p\left(x\right)-10000 \\ {Y}_1=\left(-0.05x^2+100x-2000\right)-10000 \\ {Y}_1=-0.05x^2+100x-12000 \end{gathered}$$

profit after paying tax of $10$% of profit

$$\begin{gathered} Y_2=\left(1-0.10\right)p\left(x\right) \\ Y_2=\left(1-0.10\right)\left(-0.05x^2+100x-2000\right) \\ Y_2=-0.045x^2+90x-1800 \end{gathered}$$

Plot the graph of $Y_1=-0.05x^2+100x-12000$ and$Y_2=-0.045x^2+90x-1800$

Step 3. Part (b)

According to graph of $Y_1=-0.05x^2+100x-12000$ and $Y_2=-0.045x^2+90x-1800$, Graph of $Y_2=-0.045x^2+90x-1800$ gives more profit so I would select the paying the tax of $10$% of the profit.

Step 4. Part (c)

Plot the graph of $Y_1=-0.05x^2+100x-12000, Y_2=-0.045x^2+90x-1800, and p\left(x\right)=-0.05x^2+100x-2000$

Graph of $Y_1$ is equivalent to the vertical downward shift of graph of$p\left(x\right)$ by$10000$units

Graph of $Y_2$ is equivalent to the vertical compression of the graph of $p\left(x\right)$by the factor of $0.9$

Step 5. Part (d)

Profit after paying tax of $\$4800$

Plot the graph of $Y_3=-0.05x^2+100x-6800 \& Y_2=-0.045x^2+90x-1800$

At sale of $1000$ cigars paying tax of $\$4800$ or paying tax of $10\%$ of profit, both will be same ,otherwise paying tax of $10$% of profit will give more profit.