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Problem 67 Hard Difficulty

In a certain country, income tax is assessed as follows. There is no tax on income up to 10,000 dollars. Any income over 10,000 dollars is taxed at a rate of 10%, up to an income of 20,000 dollars. Any income over 20,000 dollars is taxed at 15%.

(a) Sketch the graph of the tax rate $ R $ as a function of the income $ I $.
(b) How much tax is assessed on an income of 14,000 dollars? On 26,000 dollars?
(c) Sketch the graph of the total assessed tax $ T $ as a function of the income $ I $.

Answer

a)
b) When I=\$14,000 the tax is $\$ 400$
When $I=\$ 26,000$ the $\operatorname{tax}$ is $\$ 1,900$
c) (c) As in part (b), there is $\$ 1000$ tax assessed on $\$ 20,000$ of income, so the graph of $T$ is a line segment from (10,000,0) to (20,000,1000) The tax on $\$ 30,000$ is $\$ 2500,$ so the graph of $T$ for $x>20,000$ is
the ray with initial point (20,000,1000) that passes through (30,000,2500)

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Video Transcript

Hey, it's clear. So enumerated here. So for party, we're gonna draw a graph we have I would just income and then our for the y axis, which represents the rate of tax. So we have 10,000 20,000 and so forth. Five, 10 15 Sarah graph in red will look like this. Then there's the whole right here and so forth for part B. We're gonna do some calculations. So we have 14,000. It's greater than 10,000 but less than 20,000. So it is tasted at a rate of 10%. So we have 10% times 14,000 minus 10,000. This is 4000 times 10% and this is equal to $400 we're now gonna do 26,000 is greater than 20,000. So we're using 15%. It is 20,000 which is equal to 15% 1 6000 which is equal to $900 for part C. We're gonna draw another graft. This is income in dollars. In tax and dollars, we have 10,000 20,000 and then our graph will look something like this. No, 1000