MHB Find the tax rate t that maximizes revenue for the government?

[vaultDweller96]

Suppose that the relationship between the tax rate t on imported shoes and the total sales S (in millions of dollars) is given by the function below:
S(t) = 5 − 9 * sqrt[3]{t}

t = ?

Find the tax rate t that maximizes revenue for the government. (Round your answer to three decimal places.)
(Hint: S(t) is the total sales, NOT the government's revenue! Think about how you calculate how much sales tax you pay to the government when you make a purchase).

I'm not sure how to solve this? Do I take the derivative and set it to 0 or?

Thanks :)

 

[MarkFL]

Hello and welcome to MHB, vaultDweller96! (Wave)

First, what you need to do is define the government's revenue function, in terms of the tax rate $t$, that is, you need to determine $R(t)$. If we know total sales, and we know the tax rate, how do we compute the total revenue to the government from the sales of the shoes?

 

[vaultDweller96]

I'm a little confused sorry, how do we find R(t)? Is our total sales function the one given? Or is the tax rate the one given?

 

[MarkFL]

I'm a little confused sorry, how do we find R(t)? Is our total sales function the one given? Or is the tax rate the one given?

We are being asked to find the tax rate that maximizes the government's revenue. Now, suppose you are at the store about to make a purchase...and you know sales tax is 8%...how would you determine the amount of tax that will be charged?

 

[vaultDweller96]

We are being asked to find the tax rate that maximizes the government's revenue. Now, suppose you are at the store about to make a purchase...and you know sales tax is 8%...how would you determine the amount of tax that will be charged?

Mulitply it by the tax percetage?

 

[MarkFL]

Mulitply it by the tax percetage?

Yes! (Yes)

To find the government's revenue from the tax, and we are told to let the tax rate be $t$, and we are given the amount in sales $S(t)$, then how should we construct the revenue function $R(t)$ from these?

 

[vaultDweller96]

Yes! (Yes)

To find the government's revenue from the tax, and we are told to let the tax rate be $t$, and we are given the amount in sales $S(t)$, then how should we construct the revenue function $R(t)$ from these?

So would it be R(t) = S(t) * t ?

 

[MarkFL]

So would it be R(t) = S(t) * t ?

Yes, excellent! (Music)

So, we may state:

$$R(t)=S(t)\cdot t=\left(5-9\sqrt[3]{t}\right)t=5t-9t^{\frac{4}{3}}$$

Now, you can proceed by equating the first derivative of the revenue function to zero, to find your critical point, and then use one of the derivative tests to demonstrate that this critical value is at a relative maximum for the revenue function.

Can you proceed?

 

[vaultDweller96]

Yes, excellent! (Music)

So, we may state:

$$R(t)=S(t)\cdot t=\left(5-9\sqrt[3]{t}\right)t=5t-9t^{\frac{4}{3}}$$

Now, you can proceed by equating the first derivative of the revenue function to zero, to find your critical point, and then use one of the derivative tests to demonstrate that this critical value is at a relative maximum for the revenue function.

Can you proceed?

Yep! Thank you! I got an answer of 0.07234 after I took the derivative and set it to 0 for which my assignment said was correct. Thank you for all your help!

 

[MarkFL]

Just for the benefit of others, we find:

$$R'(t)=5-\frac{4}{3}\cdot9t^{\frac{1}{3}}=5-12t^{\frac{1}{3}}=0\implies t=\left(\frac{5}{12}\right)^3\approx7.2338\%$$

Now, to demonstrate that this critical value $t_C$ is at a relative maximum, we see that $0<R'(t)$ for $t<t_C$ and $R'(t)<0$ for $t_C<t$, and so we can then conclude that $t_C$ maximizes $R(t)$.