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If revenue flows into a company at a rate of $ f(t) = 9000 \sqrt{1 + 2t} $, where $ t $ is measured in years and $ f(t) $ is measured in dollars per year, find the total revenue obtained in the first four years.

$\$ 78000$

Applications of Integration

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Okay, So the question tells us that a company has a revenue flow that is given by FFT, which is equal to 9000 times the square root of one plus two tea dollars per year. And it wants to know the revenue in the 1st 4 years to do this. Think about it. If we wanted to do that, What we really want to do is multiply in a sense, by the years. So they cancel with this denominator. And the way we're going to do that is by integrating with respect, t because that will solve the area under this curve, which I will now draw. Here. Here is our dollars per year. And then here is our tea, which is in years. And then we have 9000 and it's a square functions that kind of diminishes like that. And that's going to be our f of tea. And so when we've taken area under this, we have this thing times We have dollars per year, times, years, we get dollars. So to do that for the 1st 4 years, we're going to take the intercourse from zero, which is presumably when the company was started to four of our function 9000 times the square of one plus two tea. Good team. And when you do that, first you're going to multiply. Are my bad divide by the derivative the inside So, too, And then we go from the 1/2 to the three halfs power. So when we divide by three halves, it's the same. This multiplying but 2/3 still multiplied by the 9000. And then we have one plus two tea, the entire factor to three halves power we want to evaluate that is your own for so clean this up. That's the coefficient. Cancel three makes the 9000 become 3000 and so we're left with 3000 times when you plug in four and a T two times for is eight plus one is nine. The square root of nine is three and then cubed. We get 27 and then we're going to subtract. When you plug in zero for tea, you get one, and then when you take one to any power, you get one. So 3000 times 27 minus one, which is 26 is going to give us final answer of 78 $1000 in the 1st 4 years