The total accumulated costs C(t) and revenues R(t) in thousands of dollars, respectively; for a coal mine satisfy:C'(t)=_t7andR' (t) = St? e-t"where ...

The total accumulated costs $C(t)$ and revenues $R(t)$ (in thousands of dollars), respectively, for a coal mine satisfy $$C^{\prime}(t)=3 \quad \text { and } \quad R^{\prime}(t)=15 e^{-0.1 t}$$ where $t$ is the number of years that the mine has been in operation. Find the useful life of the mine, to the nearest year. What is the total profit accumulated during the useful life of the mine?

Again Discussion that change in the cost that is C D S t equals two given t by 11 of a photocopy machine. Okay. And changes Revenue changing revenue, That is our destiny is given five g e raised to the power minus T square given for a photocopy machine. Okay, we are t is the time in years and revenue is in 1000 of dollars. Okay, so, first of all, we have to find out the useful life of the machine. So first part, Okay. The useful, useful life of the machine. Okay, so when the cost is equals to the revenue until then there is the useful life of machine. So we help equal C d s t equals two rd ste. OK, or we can say t by 11 is equals 25 t e raised to the power minus t square. Okay, so here t will be cut by this. And one by 55 will be erased to the power minus T square. Okay. Or we can say we will, uh, take logarithms. Um What? I've decided. So land one by 55 will be here. It will be minus T square. Okay. And now It can be written as minus Ln 55. Okay, because one by 55 can be written as 55 raised to the power minus one. And that will be minus Ellen 55. Okay. And it goes to minus T square. Or we can say Ln 55 will be t square and T will be here. Square wrote off Ellen 55. Okay. And when we saw it, then t will be okay. Two years. So this will the answer of our first part. Then the useful life of the machine will be two years. Okay. And now the second part, we have to find out the total profit accumulated during the useful life. Okay, so the profit will be the revenue minus cost. Okay. The RT minus city will be our profit. We can say the profit will be RD minus city or the integration of RDS t minus C D. S, t. Okay. And duty. This will be the total profit. And we have to find out the profit in the useful life that is 02 two. Okay, So be able to find out. Then we will be integration 0 to 2. And here R D S t that is five p e raised to the power minus t square minus c Dusty. It will be t by 11. Okay. And DT so we have to find out the profit. Okay, so, first of all this filled with the profit function, okay, For 0 to 2 years. And now we have to solve this. So it will be to solve this. Uh, we have taken here. Substitute. First of all, we will rewrite it. It will be 0 to 2. Okay, 5 80 years to develop minus T square duty minus integration 0 to 2, one by 11. Our certainty aeration. Mt. DT. OK, so to solve this in the first part, we will substitute U equals two minus T square. Okay? And it will give us the U equals two minus two T d t. Or we can say, Do you divided by minus two? It will be our t duty. Okay. And he will be okay here. We are not changing the limit. We will back. Substitute the vehicles to U equals two minus T square. Okay, so five and integration 0 to 2. And he raised to the power minus T square will be raised to the power you and this city will be. Do you divided by minus two? Okay. And minus one by 11. An integration of DDT. It will be T square by two. And the limited 02 Okay. And now five. By this minus two, it will be minus 2.5 and integration of the U. S. To the power you do. You will be raised to the power. You okay? And the limited 0 to 2 minus. It will be won by 22 T Square and 0 to 2. Okay. Here the limit is 40. And not for you. So we will back. Substitute you. That is minus T square. Okay, so it will be minus 2.5. He raised to the power minus T square and zero to minus one by 22 T Square and 0 to 2. And now we will apply the limits. So, first of all, minus 2.5. Okay. And here it will be erased to the power minus two square that is minus four. Okay. And minus. He raised to the power minus zero. Okay. Minus zero and minus one by 22. And it will be to square. That is four and minus zero. Okay. And when you saw this, it will be 2 to 7 to and this is the profit that is in the The revenue is in $1000. So it will be 1002.27 $2000. Or we can say it will be a dollar 272 Okay. And we can write it with the function of P. That is integration. 0 to 5. Okay, T he raised to the power minus T square minus one by 11 T DT. And it will be 2 to 7 to $1000. Or we can say $2272. Okay, this will be our final answer. Okay, here we have taken the use of calculator to find out the value of this. Okay. You can also use the calculator. Thank you. This will be our final answer. Thank you.

Okay, we have P primal act is a little too extreme. 30.1 part day. He in fact rightness e where? Hear me, you have to. Let's see, your CMP approximately equals zero zero euro one to one part B. What happens as a prophet? P prime x no menace Facts approaches infinity. It's gonna approach zero because using hoping tiles rule of the extra bye bye he x where it will be one device two times e of x squared, which is gonna go to zero since you're going. Since this is larger, so it will go to zero so

So here we're told that a, um, industrial machine has a decay factor of European city for per year. After six years, the value off the machine is worth $75,000 to We're trying to find the original price. So what we can do is use wth e um, exponential basic lotion. So we know that decay factor is your opens any fire, so we can use that to soften Be because, you know that B is equal to one of us are B is equal to one, uh, plus the decay. So native zero points and five senses decay. Um, you would have to put a negative. So he is basically equal to zero break 25. So which is 1/4 so we can write that, sonny. But we can also replace the Y with 75,000 because that's the final price. After six years were basically solid friend of value, A. So eight times, One over four to 86 power here. Well, because we have after six years, um, we have to solve after six years, So we have 1 46 fires a 14 times, 14 times, one for times one over four times, one over four times, one over four. So here we have 16 times, 16 times 16. So 16 times 16 is equal to 2 56 times 16 gives us 9 4006 So one to the fourth is this one over 4096. So a 75,000 is He quoted a times one over 4096. So I would consult me. I need to divide this by one over 4096. On both sides. Uh, dividing a fraction would just be most generous, appropriate. So we asked any 5000 oozing over one times 4096 over one. So this is equal to Sandy. Five times 75,000 which is equal to it's okay, let's do that again. So 75,000 times 4000 96 is equal to that d 07200000 So this is basically be 373 years and seven really in 200,000 would be the original price of

So here as to figure out the original man off a machine that had a decay factor is your foot 75 for six years and after six years, devalue us and the other 500. So the exponential function, the basic form as wise eagle to a times B to the X where a is the fence will map then be is tthe e uh, decay or growth factor in excess the number of years. So you have Wasit. Last month we have 7500 is equal to a times B. So 0.75 you know, every other decay factor times six. So, dear Pointes, every five to be six power is about a sound. 1005 100 is equal to a times zero points. And if I to this inspires what? Zero point? Uh, let's see. So we have three divided by floor to the power off six. So we have zero point once a 78 about so as we have that we can do, I vote sides by this. So we do have both sides by this. We have at a is equal to once we do that we have A is equal to 42,139 0.92 so the original amount off the machine would have been about $42,000.1 $42,189.92.