Question

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. Find the tax rate t that maximizes revenue for the government. (Round your answer to three decimal places.)

S(t) = 7 â 6(cubedroot(t))

Answer 1

66.992%

Explanation:

`Sales, S(t)=7-6\sqrt[3]{t}`

Since we want to maximize revenue for the government

Government's Revenue= Sales X Tax Rate

`R(t)=t \cdot S(t)\\R(t)=t(7-6\sqrt[3]{t})\\=7t-6t^(1+1/3)\\R(t)=7t-6t^(4/3)`

To maximize revenue, we differentiate R(t) and equate it to zero to solve for its critical points. Then we test that this critical point is a relative maximum for R(t) using the second derivative test.

Now:

`R'(t)=7-6*(4)/(3) t^(4/3-1)\\=7-8t^(1/3)`

Setting the derivative equal to zero

`7-8t^(1/3)=0\\7=8t^(1/3)\\t^(1/3)=(7)/(8) \\t=((7)/(8))^3\\t=0.66992`

Next, we determine that t=0.6692 is a relative maximum for R(t) using the second derivative test.

`R''(t)=-8*(1)/(3) t^(1/3-1)\\R''(t)=-(8)/(3) t^(-2/3)`

R''(0.6692)=-3.48 (which is negative)

Therefore, t=0.66992 is a relative maximum for R(t).

The tax rate, t that maximizes revenue for the government is:

=0.66992 X 100

t=66.992% (correct to 3 decimal places)