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.