MTH 158
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Economists use what is called a Leffer curve to predict the government revenue for tax rates from 0% to 100%. Economists agree that the end points of the curve generate 0 revenue, but disagree on the tax rate that produces the maximum revenue. Suppose an economist produces this rational function
R(x) = 10x(100-x)/15 + x, where R is revenue in millions at a tax rate of x percent.
Graph the function.
What tax rate produces the maximum revenue? What is the maximum revenue?
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I cannot figure this equation out....When I try to set it up I come up with :
10x(100-x)/15+X = R(x)
Rx = (1000x - 10x^2)/15 + x
R(x)*15 = 10x^2 + 15x
I don't think this is right can you put me on the right track
R(x) = (1000x - 10x^2)/15 + x can be written
R(x) = -10/15 x^2 + 1000 / 15 * x + x, which simplifies to
R(x) = -2/3 x^2 + 67 2/3 * x.
This is a quadratic function of form y = a x^2 + b x + c. Its graph is a parabola which, since a is negative, opens downward. Its maximum occurs at its vertex, so you just need to find the vertex of the graphs.
The zeros are easily found by factoring:
-2/3 x^2 + 67 2/3 x = -1/2 x ( x - 101 1/2),
so its zeros are at x = 0 and x = 101 1/2.
Its vertex is halfway between its zeros, which is at x = (0 + 101 1/2) / 2 = 50 3/4.
Plug this x value into the function and you will get the maximum revenue.
Thanks
Billy
Good question.
&#See my notes and let me know if you have questions. &#