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Store sells Christmas lights. Suppose that the store pays Sl to purchase set of lights: They sell the set for $2. Any set not sold by Christmas will be discounted ...

Question

Store sells Christmas lights. Suppose that the store pays Sl to purchase set of lights: They sell the set for $2. Any set not sold by Christmas will be discounted to $0.5 and sold at this price. Suppose that the probability mass function for the demand D of the Christmas lights is given as: P{D 500} 0.1, P{D 600} 0.2, P{D 700} 0.1 , P{D = 800} = 03, P{D = 900} 0.2_ and P{D = 1000} = 0.1What is the expected demand? What is the expected profit if the store purchases 800 sets of lights? What is th

store sells Christmas lights. Suppose that the store pays Sl to purchase set of lights: They sell the set for $2. Any set not sold by Christmas will be discounted to $0.5 and sold at this price. Suppose that the probability mass function for the demand D of the Christmas lights is given as: P{D 500} 0.1, P{D 600} 0.2, P{D 700} 0.1 , P{D = 800} = 03, P{D = 900} 0.2_ and P{D = 1000} = 0.1 What is the expected demand? What is the expected profit if the store purchases 800 sets of lights? What is the optimal number of lights should the store purchase to maximize its expeeted profit? What is the maximum expected profit?



Answers

The cost and price-demand functions are given for different scenarios. For each scenario, $\bullet$ Find the profit function $P(x)$ $\bullet$ Find the number of items which need to be sold in order to maximize profit. $\bullet$ Find the maximum profit. $\bullet$ Find the price to charge per item in order to maximize profit. $\bullet$ Find and interpret break-even points. The monthly cost, in hundreds of dollars, to produce $x$ custom built electric scooters is $C(x)=20 x+1000, x \geq 0$ and the price-demand function, in hundreds of dollars per scooter, is $p(x)=140-2 x, 0 \leq x \leq 70$.

Alright. Working with applications of linear functions, we often want to find profit revenue cost functions to build a graph and find their break. Even points in this case were given the information that affirm is able to turn a profit of $4500 when they produce six produce and sell 600 items. But they can earn a profit of $10,500 by selling 10,000 items and that they're selling these items at a price of $2.25. To begin with this information to find our cost curve, we're gonna first identify our profit curve from here. We want to remember that profit is equal to revenue minus cost, and we can start to fill in the variables that we do have. So from here we can say that profit is equal to 2.25 which is our price Times X minus D minus M x. I just distributed this negative here into their from here we can factor out our X just to simplify a little bit. So we get P is equal to X times 2.25 minus m my honesty and moving on. What we can do is we want to now substitute the information that was given to us into this profit function. We know that we have a profit of $4500 associated with an X value of 6000 plugging all of this in We get that and then we want to do the same thing with the other. Information that was given to us at a profit of 10,500 is equal to an X value of 10,000 2.25 minus M minus D. We now want to subtract each of these functions from one another, giving us 6000. It's equal to 4000 times 2.25 minus m. With that, we then just simplify so that we can solve for em. That gives us 1.5 is equal to 2.25 minus m. I got the by dividing both sides by 4000, and from here we're able to arrive at the conclusion that em is equal to 0.75 We'll need that for a cost function civil box set in so we can't lose it next weekend. Take the functions that we were using up here and we're going to and put the new information that we have, which is the value of em to give us 4500 is equal to 6000 2.25 minus RM value of 0.75 minus D. This allows us to solve for D because the only variable in the equation we then get 4500 is equal to 6000 times 1.5 minus D. From that, we can say that 4500 is equal to 9000, which was 6000 times 1.5 minus D and were able to arrive at D is equal to 4500. It's another piece of information for a cost function toolbox that in with that, we're able to input those variables into a cost function. To get CFX is equal to 4500 plus 0.75 x, and that is our cost function. Next, if we want to find what our break even point is, we want to just recall that a break even point is the point at which our cost function and our revenue functions intercept one another. We already know what our cost function and our revenue functions are. So we just need to set them equal to each other. Cost function, as we just said, 4500 plus 0.75 x setting that equal to a revenue function which is 2.25 times x solving for X 4500. It's equal to 1.5 x, and we get X is equal to 3000. Is our break even point If we wanted to depict these curves graphically conceit Up here we have a graph sketched out. We know from our cost function that our intercept is 4500 Would be this point over here and we just found that a break even point is 3000 that sits right here. And we could plot that point. If we were to draw lines through each of these, we would get the exact picture that we have right there. And now supposing that we have a that this firm now has a profit equal to $90,000. They have an X value equal to $30,000 or 30,000 units and they want to know at what price they should sell these items in order to earn that profit of $90,000 we can recall once again that our profit is equal to revenue minus cost. Doing this, we can put the information that we have of $90,000. And we're really just using one of these equations, pretty much replicating this equation here, but with new values. So we have 7 90,045 100 which is equal to our new X value of 30,000 times peaks. We don't know what our prices minus 4500 plus 0.75 times 30,000 solving this we have 90,000. It is equal to 30,000 p minus 27,000, adding 27,000 on both sides. We get 117,000 is equal to 30,000 p and then solving for P. We end up getting, but the price is equal to $3.90. That is the price at which they should sell their products in order to earn a profit of $90,000

Hi there for a problem that we need to hold too late First the demand function. And we are given two points that 1000 in 200 tablets, computers. Our sell by $350. And also that if we and sell um an additional of 80 tablets. So this will be dance. We can sell that for $10 less. So that will be $340. So this is the given information and we need to find the demand function. So ah we will set that P. Is the price and Q. Is the number of tablets sold. So that the function of price should be something like this. Where m. Is the slope of dysfunction. And P zero corresponds to a given point. And And Q002 another given point for the number of top of that let sold. So to find first the slope we know that that is change and in price over the change in the number of chocolate salt. So we can see in here that we put the final price by the initial price over. Then find a number minus the initial number. So the final price or the second point in here is given by this because the p um the B component or the price is greater. That so that will be the final the final part. Oh no sorry. This is the price. This is the christ. This is the initial um number of tablets. Uh huh. This is the final one. This is the initial price and this is the final price. So with that set which just simply substitute those. So it will be 340 -350 over mm So this will give us -1/8. And this is the slope. So we just substitute this slope in this equation and we put them some values for piece of C. Rank is zero. I'm going to use this pointing here So we will have pete minus piece of zero which is 350 is equal to the slow times cute minus 1000 and 200. Yeah. So solving for P we will have that function is 1/8 you loss 500. And this is the demand function. No for part B of this problem. And is what should the price be said in order to maximize the revenue. So in this case we need to find the revenue function. Um They divided fined the maximum and then substitute that value in the price function. That's what we need to do for power bi. So the revenue function. Okay that will come art of Q. Is the price function or the demand function times the number of have let sold. So that if we multiply that function bike you will have 1/8. Q square plus by 100 Q. And to find a critical point. We I don't know if it is a minimum or a maximum for to find a critical point. We divide this function with respect to Q. So we will have 1/4 cube glass. 500. And to find a critical point. We said this to see rope. So it is very straightforward that will obtain Q. is equal to 200. So this is our a critical point. So to know if this critical point is a maximum or a minimum, we need to do A second derivative of the revenue function with respect to queue. So that will obtain -1/4. And this value is less than zero. So that we will have a maximum At you over 200. And with that we just think this is to do that value in the function of the price. So it will be and is functioning here. They were obtained for the previous eaten. So we will have 1 1/8. Two trends blast Okay and if we block this into the calculator we obtain $250. No for parse theme of this problem we need to oh um we are given some cost function and we are asked that what price should it choose in order to maximize its profit. So function that we are given is Yeah that's and in this case as represent the number of of tablets sold. So it is equal to Q. And to obtain the profit. The function of profit is equal to the revenue minus the cast function. So from the previous we'll have the revenue function and so that will be that the prophet function of ads in this case we'll kind of do it with ads but we you can do it also with Q. It doesn't matter. So it will be one over eight. That's a squirt plus 500 adds minus the gas function that we are given. So it will be mine is 35,000. Uh huh. I'm sorry this is -2 because all of the cows function um should be multiplied by the miners. So this will be test um We can simplify tests. We will have one over feet squared. Um This will be to us 380 ask mind is this so again we want to maximize this function and then with that value that we obtained we substituted in the function for the price. Um so that we will maximize the profit. So again to find a critical point of the function we in the rebate with respect to the variable X. In this case we will have 1/4 at last. 380. Okay. And we set this to zero the hood because we want a critical point so that we will find that adds yes 1000 and fight 120 to know if this value is um is a maximum or a minimum. We need to revive these two times and again we'll obtain that this is minus 1/4. This value is less than zero. So we will have a maximum But at equals two the value that we just have obtained. And we just simply need this institute that value in the price function. So the price function remember is the first one that week obtained this in here. Q. Is equal to add. Remember that? So dot We will have the price that makes the maximize that profit is equal to minus 1/8. The value that we obtain for ads which is this um blast by hundreds and this give us $310. So these And or charging $310 per tablet will not see mind the profit. So thank you so much and this is it for this problem.

The revenue function and the cost function for producing X number of Candies is given us are off. X is equal toe 9.5 x minus 0.4 x squared on see if X is equal to 1.25 x plus 250 now we need to calculate the value off the Candies where the revenue function will be maximum. So that will be the X coordinate of the vortex of the revenue function, which will be at minus be divided by two way. Now comparing the given revenue function with the standard form, this will be equal toe minus 9.5, divided by two multiplied with minus 0.4 which will be equal toe 118.75 which is approximately equal 219 Candies. So the revenue function, 419 Candies will be the maximum, which is equal to 9.5 multiplied with 119 minus 0.4 multiplied with 119 hold square, which is equal to $564.6. Next, we need to find the prophet function, which has given US revenue function minus the cost function, which is equal to minus 0.4 x squared plus 8.25 x minus 250. Now we need to find the number of watches which have to be produced to maximize the profit that will be again at the X coordinate of the vortex off the prophet function. Now that will be at again minus be divided by two way. Now comparing the prophet function with the standard form, this will become equal toe minus 8.25 divided by two multiplied with minus 0.4 which will be equal toe 103 candy. So the profit 403 Candies will be equal toe minus 0.4 multiplied with 103 hold square plus 8.25 multiplied with 103 minus 250 which will be equal to $175.4. No for the answers in a and C part. There is a difference because a part describes when we can maximize the revenue. While see part describes when we can maximize the profit now profit is the difference between two functions. While revenue is straight away, a function now revenue function is a quadratic function. Andrea, using a quadratic function to represent the revenue because off the reason now revenue function, which is dead. It represents the production off Candies, taking into consideration the whole lot, many factors. So that is why a revenue function is more or less a quadratic function.

For the following problem, we have the demand price demand function for this television that's going to be Kiev X is equal to negative zero point. Mhm. There are five x plus 600. And we know that the revenue is going to be key. FX uh locate the annex index. It's not the revenue function. And then we know the cost function is given to us As 0.000 02, ask cube zero or 0.03 x squared. And then plus 400 X plus Mhm. The government. Yeah. So then we're gonna have our profit function, which is equal to the revenue function minus the cost function. And that's going to end up allowing us to maximize profit um right here for our final answer.


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