Science:Math Exam Resources/Courses/MATH100 C/December 2024/Question 12 (c)

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MATH100 C December 2024

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Question 12 (c)

Consider an income tax system in which the tax $T$ paid on an income $x$ is given by the formula $$T(x)=\frac{x^3}{1+2x^2}$$ where both $x$ and $T$ are measured in units of $\mathrm{\$}10,000.$ It is given that $$T^{\prime }(x)=\frac{x^2(2x^2+3)}{(1+2x^2{)}^2},\text{and},T^{″}(x)=\frac{2x(3-2x^2)}{(1+2x^2{)}^3}.$$ Approximately what fraction of their income is paid as tax by people with very large incomes?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Consider $\frac{T(x)}{x}$ and compute its limit as $x\to \mathrm{\infty }$.

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Solution

We consider the fraction of income paid as tax: $$\frac{T(x)}{x}.$$ Compute the limit as $x\to \mathrm{\infty }$: $$\underset{x\to \mathrm{\infty }}{\lim }\frac{T(x)}{x}=\underset{x\to \mathrm{\infty }}{\lim }\frac{x^3}{x(1+2x^2)}=\underset{x\to \mathrm{\infty }}{\lim }\frac{x^2}{1+2x^2}.$$ Divide numerator and denominator by $x^2$: $$\underset{x\to \mathrm{\infty }}{\lim }\frac{1}{\frac{1}{x^2}+2}=\frac{1}{2}.$$ Thus, for very large incomes, approximately $\frac{1}{2}$ of income is paid as tax.

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