SOLUTION: In Fantasy Island an individual pays 10% tax on gross annual income not exceeding $200,000. For every dollar on the next $300,000 the tax is 15%. Any income in excess of this is

Algebra ->  Functions -> SOLUTION: In Fantasy Island an individual pays 10% tax on gross annual income not exceeding $200,000. For every dollar on the next $300,000 the tax is 15%. Any income in excess of this is       Log On


   

Question 884013: In Fantasy Island an individual pays 10% tax on gross annual income not exceeding $200,000. For every dollar on the next $300,000 the tax is 15%. Any income in excess of this is taxed at 20 cents in the dollar.
Using T to represent annual tax liability and x gross annual income:
a. Determine the annual tax liability function T(x) for an income earner in Fantasy Island.
b. What is the tax liability of a person who earns a gross annual income of $350,000?
c. What is the tax liability of a person who earns a gross annual income of $500,000?

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
very tricky function and i just figured it out after many previous tries where i failed.

you are the beneficiary of this sudden breakthrough in knowledge.

previously i thought it had to be a piecewise function, which it can be.

today i learned that it doesn't have to be, and can be modeled with one continuous function.

i'm not sure if you're supposed to be sophisticated enough to figure this out, but i'll give you the solution and you can certainly use it if you like.

If you can't use it, or don't want to, then you'll need to resort to a piecewise function or figure out another function on your own.

the function that i just discovered today is shown in the following pictures along with examples of how it would be used.

the formula uses (x-0) + |x-0| and (x-100) + |x-200| and (x-500) + |x-500|

the expression |...| means absolute value of (...).

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the graph of the equation looks like this:



you can confirm the solution is accurate by using the folloiwng logic.

x = 200
tax = .10 * 200 = 20.

x = 350
tax = .10 * 200 + .15 * 150 = 20 + 22.5 = 42.5

x = 500
tax = .10 * 200 + .15 * 300 = 20 + 45 = 65

x = 800
tax = .10 * 200 + .15 * .15 * 300 + .20 * 300 = 20 + 45 + 60 = 125

this is because:

you pay 10% for the amount above 0 and less than or equal to 200.
you pay 15% for the amount above 200 and less than or equal to 500.
you pay 20% for the amount over 500.

the expression works because:

when x is smaller than y, the expression (x-y) becomes -z and the expression |x-y| becomes + z and the -z and the +z cancel each other out resulting in 0.

when x is larger than y, the expression (x-y) becomes z and the expression |x-y| becomes z and you get 2z which is then equal to z after you divide it by 2.