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Write for me business research topics thesis kapstone paper and packaging locations for money dms iitd placement report example what's up everybody so in this question we're gonna be dealing with a word problem that relates to piecewise functions so it's a word problem that you've probably seen a few times so it's a tax word problems so under a certain tax law the first 60,000 of earnings is subject to 30% tax earnings greater than 60,000 are subject to 42% tax and there's two parts to the question part a we have to write a function that models this situation and then Part B we have to determine the tax on earnings of a hundred and ten thousand so to make a function for this situation let's first show what's going on in a diagram just so we can maybe visually see it better so there's actually no limit here this keeps going on forever so this is earnings of zero so anything up to $60,000 in earnings what's happening is you are being taxed on that earnings at 30% so for example if you make twenty thousand then you're getting taxed thirty percent that would be six thousand dollars if you're making forty thousand thirty percent of that is twelve thousand so that's your tax twelve thousand dollars on forty thousand dollars worth of earnings if you're making sixty thousand thirty percent tax on that is eighteen thousand so anything from sixty thousand and below the tax is thirty percent but what happens is anything after sixty thousand anything greater than sixty thousand is subject to forty two percent tax now here is where people get a little confused whatever's over sixty thousand that full amount doesn't get taxed only the portion that's above sixty thousand so for example let's say that you make 80 thousand well what's going to happen is that first sixty thousand is going to be taxed at thirty percent and then that remaining twenty thousand the amount over sixty thousand right 80 thousand minus sixty thousand twenty his tax I have 42 percent all right so that's just an example so knowing that how can we make a function for this situation or for any situation for any earnings that we input in the function we could figure out what the tax is right away well let's introduce some variables here so let this capital T equal tax paid and then let X equal your earnings so what we're going to do is we are going to make a function for the tax paid in terms of the earnings so depending on whatever the earnings are we plug in the earnings into our function and we'll get the tax so notice how this function there's going to be two cases for it so the first case is if the earnings are less than or equal to 60,000 right so that's going to be this case here so that's going to be the first case and then the other case is if earnings are greater than 60,000 well as we mentioned if earnings are less than or equal to six and thousand all you're doing is you're taking that earnings and you're taking 30% of that so it would just be point three X remember you got to change this percentage into a decimal so 0.3 X so again if you're making 40,000 if we plug in 40,000 for the x point three times 40000 gives us 12,000 so that would be the tax if our earnings was 40,000 now what about if the earnings are greater than 60,000 so let's take that same example we talked about the 80,000 and let's just pretend it's some value X okay so we don't know what the earnings are but let's say a general value we just know that X is going to be greater than 60,000 well what's the tax going to be when X is greater than 60,000 well if it's greater than 60,000 then we for sure know that for the first 60,000 we have to pay this 30 percent tax so we know that our tax is going to be 60,000 times the tax rate of 0.3 write that first 60,000 we're definitely paying 30 percent tax on that and then what's going to happen kind of running out of room here so I'll just continue it down here is that the portion of the earnings above 60,000 so let's represent that as X minus 60,000 right so the portion of the earnings above 60,000 which we can represent as X minus 60,000 so for example if this is 80,000 if we go back to that example this bracket here X minus 16 thousand would be 20,000 80,000 minus 60,000 so that portion above 60,000 is getting taxed at 0.42 or 42% so now we have this expression 60,000 times 0.3 plus X minus 16,000 in brackets times point 4 2 so I took this expression and rewrote it in one line so I got 60,000 times 0.3 plus I brought this point 4 2 over in front of the bracket I got the point 4 2 in front of X minus 60,000 so let's simplify this a little more so 16,000 times 0.3 that gives us 18,000 which is the tax we're going to have to pay on that first bracket Plus this point 4 2 I'm going to distribute inside the brackets 0.42 times X gives us point 4 2 X and then point 4 2 times negative 60,000 that gives us negative twenty-five thousand two hundred and to simplify this more notice how this 18,000 and this negative twenty-five thousand two hundred those are like terms eighteen thousand minus twenty five thousand two hundred that gives us negative seven thousand two hundred so let's put this point for 2x in front and then 18,000 minus twenty five two hundred would give us minus seven thousand two hundred and that is the simplified function for the tax if the earnings are greater than sixty thousand so that expression you would put here and you are done Part A so this is the simplified function for the tax paid and notice how there are two cases when X is when the earnings are less than or equal to sixty thousand and when the earnings are greater than sixty thousand so moving on to Part B we have our function from Part A and now we have to find the tax when the earnings equal a hundred and ten thousand well if earnings are one hundred and ten thousand which case are we dealing with the first case or the second case well the second case because the earnings are greater than sixty thousand when the earnings are greater than 60 thousand the tax paid it's going to be this function here so to find the tax actually let's do it over here to find the tax when the earnings are one hundred and ten thousand you plug it into this formula so we'd have 0.42 times a hundred and ten thousand minus seven thousand two hundred so point four two times one hundred and ten thousand that would give us forty six thousand two hundred minus seven thousand two hundred and then 46 200 - 7,200 that would give us 39,000 so that's the answer to Part B so if you're making a hundred and ten thousand dollars you got to pay 39 thousand dollars in tax and you can check your answer with that diagram that we made before right if you remember so sixty thousand thirty percent anything over sixty thousand is 42 percent and we're making a hundred and ten thousand so the first 60 thousand that's tax at 30 percent and then a hundred the amount over sixty thousand to get one hundred ten thousand is fifty thousand so fifty thousand of that one hundred and ten thousand is taxed at forty two percent so this would end up being eighteen thousand if we multiply these two and then if you multiply these two you would get twenty one thousand and then when you add these you get thirty nine thousand and notice how that's the same answer that we got here so this answer makes intuitive sense so we can be pretty confident that our function is correct and we got the correct tax on a hundred and ten thousand dollars worth of earnings yo what's up guys thanks for checking on my channel hopefully you got some value from the video you just watch if you did get some value a big favor to ask you please like this video and subscribe to the channel any questions any recommendations on things you'd like to see please leave it in the comment section also check out the description box below for links to material and content related to the video you just watch peace out road to implementation enterprise risk management for colleges and universities Metropolitan College of New York.