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Company detorminos tnat its marginal rovonto per day given by R'(t) - 110 2 R(O) = where R(t) the rovenue in dollars, on the tth day: The companys margina cost...

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Company detorminos tnat its marginal rovonto per day given by R'(t) - 110 2 R(O) = where R(t) the rovenue in dollars, on the tth day: The companys margina cost per day given C' () = 90 _ 0.3t, C(O) = 0, where C(t) = the cost, in dollars, on the (" day: Find the total profit from to [ = 8 (the first days) Round to the nearest dollar:Note: P(T) = R(T) - C(T) = [R"() - C'(t)] at

company detorminos tnat its marginal rovonto per day given by R'(t) - 110 2 R(O) = where R(t) the rovenue in dollars, on the tth day: The companys margina cost per day given C' () = 90 _ 0.3t, C(O) = 0, where C(t) = the cost, in dollars, on the (" day: Find the total profit from to [ = 8 (the first days) Round to the nearest dollar: Note: P(T) = R(T) - C(T) = [R"() - C'(t)] at



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Great Green, $\ln c .$ determines that its marginal revenue per day is given by $$ R^{\prime}(t)=75 e^{t}-2 t, \quad R(0)=0 $$ where $R(t)$ is the total accumulated revenue, in dollars, on the tth day. The company's marginal cost per day is given by $$ C^{\prime}(t)=75-3 t, \quad C(0)=0 $$ where $C(t)$ is the total accumulated cost, in dollars, on the tth day. a) Find the total profit from $t=0$ to $t=10$ (see lixercise 49 ). b) Find the average daily profit for the first 10 days.

In Brabant 49. We have the marginal revenue birthday for a company given by this equation and knowing that the starting day the revenue is zero and we have the company's marginal cost per day given by this equation, knowing that the cost at day zero equals zero, we want to calculate the total profit from time zero tow time. 10 for the 1st 10 days, full body for body. We want to get total profit. The total profit schools the revenue minus costs. But we have the per day, the birthday revenue and the per day cost. Then we can get our by calculating the definite integral from 0 to 10 for our dash off T d t minus. We can get the cost to buy calculating the different integral from 0 to 10 or dish. See this off T DT. Let's evaluate these two Integral. We can take a common the girl from 0 to 10 for the function or 100. It is a war of T minus C, which is 100 minus. Oh, point duty DT The evaluation off 100 Eat the war of tea. The integration is 100 off multiplied by eats a lot of tea but employed by the differentiation of tea minus divided by the differentiation of tea, which is one minus 180 plus all point to t squared, divided by two and we integrate from 0 to 10 equals. We substitute first by the hour bond he so sued by t equals 10 100 not to blow it by e the war often minus e multiply beytin plus 4.1 multiply it by 10 squared, which is 100 minus. We substitute by t equals zero. We have 100 multiplied by it is a war of zero, which is one minus zero equals 100. Deployed by E was about often minus 100 multiplied by 10 loss 10 minus 100. It equals two million, two million and 201 thousands and 500 and 50 seven. This is for the total profit for the first 10 days for bar TV, we want to calculate the average daily profit for the 1st 10 days. We can calculate the average their profit average be average profit average over that satisfied interval equals the definite integral oh, or minus C DT off our dish minus C d d t from 0 to 10 divided by the length off. The interval, which is then minus zero, equals one divided by 10 multiplied boy, The definite integral off or dish which is 100. It was a lot of tea minus deep minus or which is 100 minus. Oh boy Duty DT, Integrate from 0 to 10 equals Oh boy Tom, multiply it, boy. The integration of 100 m or of tea We have integrated this term already embody. Then we can multiply. Boy, this number toe millions 201 thousands 50 557 equals 22 3200 hundreds, 2200 thousands and one 155. This is the average profit through the 1st 10 days on the final answer off part B and this is the final answer off party

We're as to find the the amount of profit for a given company. And so how the profit is found is by finding how much you make on a daily basis, minus how much you have to spend on a daily basis. And so the the amount of product 100 minus 1000.1 Q times the product times the amount of product sold is our is our daily revenue, So Q is thea amount of product times the price of the product 100 minus 1000.1. Q is equal to the the daily income, and so that could be simplified to 100 q minus 1000.1 q squared, and that is equal to our revenue. And so, for the cost function, it costs $10,000 to operate every day. And you also have to pay an additional $12 for every every product, every unit that that we produce. And so when we subtract this, we get negative. 0.1 Q squared plus 100 que minus 12 q minus 10,000 and we want that to be greater than or equal to zero to see where we make even and between what values, how many How many products do we have to sell to keep to keep our profits positive? And so this could be simplified to negative 0.1 Q squared plus 88 Q minus 10,000. And so if we were to solve this by using by using the quadratic formula, we would get negative 88 negative. I gotta be plus or minus the square root of B squared, which is 7744 minus four times and get a 4.1 minus 4000 minus. Sorry, this is four times negative 40.1 times 10,000 times negative 10,000 which gives us negative 4000. So this is for times. This is four times negative 40.1 times 10,000 and so that's attracted and it's all over. Two times are 82 times negative 820.1, and so when we solved when we saw this, this quadratic formula we are left with Q is equal to 1 34.5 and queues equal to 7 45.9 And so we can't create we can't make. We can't make 0.9 of a product, so we have to. We have to round down. So so if Q So if Q let's OK, let's say let's say we have to positive values and and we know that our A value is negative 0.1 and therefore therefore a is less is less then zero. We know that we will have a curve that looks something like this. And so this is the part that we're focusing on. This is our positive. And so this is the region where we will make a profit. And so since we can't make, we can't go at 7 46 and at 1 35 we know that we would not are at at 1 34 So this is 1 34 25 and this is 7 45.9 This is this is seven 45 20 nine. He's two intercepts. And so we know that if we go less than 1 35 so we cannot go, we can't have 1 30 We can't have 1 34 We also can't have 7 46 So we know that our Q has to be bounded by 22 integer numbers, so Q has to be less than or equal to less than or equal to 7 45 since that's the biggest hole number we can make. And it also has to be bounded by. It also has to be bounded by 1 35 one 35. Since we we technically, we can't make 1 34 Otherwise we'd be at a negative. So our biggest hole number, uh, problem or our smallest whole number for profit is 1 35 units and are our smallest. Our largest hole, number four profit is 745 because if we go even a little bit further, let's say 7 46 where we're at a deficit. And if we go a little bit smaller than 1 35 let's say 1 34 We're at a deficit, so we want to operate within this range. We want to sell between 135 and 745 units off product

In this question were given the marginal cost function. See, Graham. Thanks equals 232 plus x, our 20 and want to find the increasing in the cost form the level off 15 to the 20. So it means that what you need to find that in the girl from 15 to the 20 on the Supreme X, t X. Then we get equal to the integral from 15 to 20. 32 plus Exxon 20 the X. Yeah, Andi. And entirely reverting the 30. Joey, could you through the two x entirely revert him? This one? You go. Did you one of the 20 times X square? Over to on now evaluated from 15 to the 20. Now, if we put the 20 inside, we have it will be 640 and then plus, here we have the 20. We have to be defined. Invite You will be 10 and then minus It would would have fit in in time. We have to touch it two times 15 and then for the foundry 80 plus This one will be 15 square defending by the 40 have encouraged your 5.6 25 and our own get me go to plus for anything and then we have the 6 50 my understand. So I'm going to go to 164.3, $75 and that's maybe the additional cost him.

A company that manufactures travel clocks, has determined that the daily Marginal Coast function associated with producing T's clocks is derivative of C of X equals 0.9 times X square, minus 0.9 times X plus eight we're seeing Derivative of X is measure in dollars per unit, and X denotes the number of units produced. Management has also determined that the daily fixed coast incurring producing these clocks is $120. And with that information, we want to find the total cost incurred by the company in producing the first 500 trouble clocks per day. Right so way have the marginal cost function that is the derivative of the cost function. We want to find the glucose. So we got to find the until derivative or indefinite integral off. It's c prime. So we know that total cost see of X will be equal to the integral off the marginal coast. Uh huh. What? And that will be equal to the integral. The indefinite integral off the expression given to Thean marginal coast that is zero point 000 seriously or nine x square, then minus 0.9 x plus eight differential ipix. And we got to integrate this and this is a constant. So we get 0.9 times interval of X Square minus 0.9 which is a constant times the integral of X plus eight Internal of Differential of X And these are powers of acts and we know how to integrate that. It's 0.0 00 Sierra nine x to the third over three minus 0.9 times X squared over two plus eight eggs, plus a constant of integration. We put the continent's constant of integration at the end because even if each integral has a constant of integration, we know that some of them will be a unique constant of integration. So the tractor off being indefinite is the reason why we have constant of integration, which has to be determined with some initial condition. Let's simplify a little bit here, so we have 0.0 9/3 0 point 000003 X cube minus 0.9 Number two is 0.0 45 x square plus eight x plus c and this is the total cost function. With it, we can calculate it'll cost at any for any number of units. But before we get to determine this value, the value of this constant of integration. And for that we have the initial value given here. The management determined that daily fixed coast in current position disclose clocks is $120 the people. So because see of zero is 120 the fixed cost of position the unit, it's $120. I remember this function is given in dollars. It's important because the marginal coast is given in dollars per unit. So the coast, because it will be given in dollars so see of Siri's is 120. If we relate this expression at zero X equals zero, we get C. So so you gotta be 120. Okay, then The total cost function is 0.3 X cube minus 0.0 45 X square, plus a X plus 120 in dollars. And now we are able to calculate. What the problem is asking for is to find the total cost incurred by the company in producing the 1st 505 o'clock per day. So total cause mhm off producing to first. Okay, 500 clocks or traffic clock ace. See, at 500 that is 0.3 times 500 to the third, minus 0.0 45 500 square. Right. Plus eight times 500 plus 120. And using a calculator, find disease equal to 3000 370. So okay, during coast Uh huh. Incurred by the company? Yeah. In producing the first in producing the first 500 travel clocks per day is three 1003 $170. And that's do you final answer off given probe.


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