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Tax Formula

Here's the problem I'm having trouble with:

Taxes in Oz are calculated according to the formula T=.01I^2 where T=thousand of dollars of tax liability and I=income measured in thousands of dollars.

How much in taxes is paid by individuals with incomes of $10,000, $30,000, and $50,000? What are the average tax rates for these income levels? At what income level does tax liability equal total income?

Marginal tax rates in Oz can be estimated more precisely by calculating tax owed if persons with the incomes in part a get one more dollar. Make this computation for these three income levels. Compare your results to those obtained from the
calculus-based result that, for the Oz tax function, its slope is given by .02I.

I have figured that the taxes are 1,000, 9,000 and 25,000 and that at 100,000 tax liability equal total income. My difficulty is in calculating the marginal tax rates and comparing to the slope of .02I.

Solution Preview

Here's the problem I'm having trouble with:

Taxes in Oz are calculated according to the formula T=.01I^2 where T=thousand of dollars of tax liability and I=income measured in thousands of dollars.

How much in taxes is paid by individuals with incomes of $10,000, $30,000, and $50,000? What are the average tax rates for these income levels?

T = 0.01I^2

Tax liability for income of $10,000 (I = 10):

T = 0.01(10)^2
T = 0.01(100)
T = 1

Tax liability is $1,000. Tax rate is $1,000/$10,000 = 0.1 = 10%.

Tax liability for income of $30,000 (I = 30):

T = 0.01(30)^2
T = 0.01(900)
T = 9

Tax liability is $9,000. Tax ...

$2.19