Question

Under Antarctican tax laws, your first $100,000 of income is tax free. However, after that, your tax rate increases by 5% for every $10,000 additional dollars you earn (for example, if you make between $120,000 and $129,999.99, you will be taxed 15% of your total income). What is the optimal amount of pre-tax income (the least pre-tax amount earned which yields the most take-home pay, rounded to the nearest cent)?

Answer 1

Choices:
A) $149999.99
B) $159999.99
C) $129999.99
D) $269999.99
E) NOTA
Can you explain to me?

Answer 2

f(x) = x - 0.05x((x - 100000)/10000)
Does that make sense for this problem? I can just find the highest point on the graph (the vertex) and that tells me how many 10000 extra there should be?

Answer 3

Would it be easier to write the tax function and minimize it?

Answer 4

Maybe? T(x) = (100000 + 10000x)(1 - 0.05x) which equals -500x^2 + 5000x + 100000? Like that?

Answer 5

Well maybe I don't understand the problem but isn't the tax rate variable?

Answer 6

Well, it says that it's 15% if you have a surplus of $20000 (5% extra for each $10000). THat means that the original tac is rate is 5% because 15 - 10 = 5?

Answer 7

I think it means that if you make 100,001 your tax will be .05(100,001)
If you make 110,000 your tax will be .10(110,000)
If you make 120,000 your tax will be .15(120,000)

Answer 8

Ok? I thought that's what i said before. THat would give you the tax if the original tax rate without the surplus is 5% right?

Answer 9

If there is no surplus, there is no tax.

Answer 10

If you make 100,000 your tax is .00(100,000)

Answer 11

Oh ok. The people that made the problem gave an explanation, but I didn't understand it that well, so I came here. Would you like to see it?

Answer 12

Yes.

Answer 13

Yes. That makes sense. They based it on take home pay. And they determined the function for even multiples of 10,000 and then checked the endpoints of the intervals. If you think of the tax rate in relation to income, it's like a piecewise function or a greatest integer function.

Answer 14

So I guess the simplest way to find the answer would have been to take the choices and calculate the tax on each income amount, subtract it from the income and choose the biggest answer.
For example, the tax on 149,999.99 would be .25(149,999.99) so the take take home pay would be .75(149,999.99)=11249.99
On 159999.99 you would take home .7(159999.99) =111999.99
And so on.
Do you understand?

Answer 15

Yeah. I understand it now :). I guess the reason I didn't do that is because there was always the possibility that the answer wasn't there, hence E NOTA. Thanks you!

Answer 16

yw