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When wholesaler sold product at $35 per unit, sales were 250 units per week After price increase of $7_ however; the average number of units sold dropped to 230 per...

Question

When wholesaler sold product at $35 per unit, sales were 250 units per week After price increase of $7_ however; the average number of units sold dropped to 230 per week. Assuming that the demand function is linear; what price per unit will yield maximum total revenue?

When wholesaler sold product at $35 per unit, sales were 250 units per week After price increase of $7_ however; the average number of units sold dropped to 230 per week. Assuming that the demand function is linear; what price per unit will yield maximum total revenue?



Answers

Maximum Revenue When a wholesaler sold a product at $\$ 40$ per unit, sales were 300 units per week. After a price increase of $\$ 5,$ however, the average number of units sold dropped to 275 per week. Assuming that the demand function is linear, what price per unit will yield a maximum total revenue?

So in this problem were given a sporting goods wholesaler finds that when they sell a certain product at $25 they sell 500 units and when they so it at $30 it sells for 460 years. So first we want to find we're gonna call this P for price and Q for demand prices and dollars. Demand is a units. So if I want to write demand as a function of P and I assume that this is a linear graph and we can write it as so the difference in quantity, which is gonna be for 60 minus 500 so it will be negative 40 over the difference in price, which is $5 times p minus 25 plus. So at when we plug in 25 we expect 500. So we get zero sum. He's only 500. So now I can distribute the negative eight to get negative eight p plus 200 plus 500 is equal in the wall. Have negative eight p. Yeah, plus 700 is equal to my cure you. So another problem wants us to find revenue. So revenue of key is going to equal the demand with respect to P times P. So that's gonna equal negative eight p squared plus 700 p. So now that we have the revenue, we want a plot that that's gonna look like this. So we have an intercept. Ah, p equals zero because we're making $0 times, whatever the demand is, and then which would be 700 units, then we have another intercept over here. So we have to figure out what that ISS So we'll have negative eight p plus 700 is equal to zero. So that means P is gonna equal 700 over eight. So what's that gonna be? Uh, we're gonna have so 3 50/4 What? 75/2. And then finally. So it's you 0.5, whatever that that is 85 get us 87.5 and it's a proble that's facing down. Assume this was symmetric. So now if we want to find this maximum value, we know that it's gonna be the average of our two zeros because the problem is symmetric across its vertex. So if you want to find that value, we can take 87.5. Let's Zira over to which will get US 43.75 That's in dollars, because it's on the P access in the prices and dollars are all right. And we want to know what the revenue is of this price. So it would take are of 43.75 which is gonna be equal. Teoh Negative eight times 43.75 squared plus 700 times 43.75 So that gets us 15,000 $312.50 as the maximum revenue for this product.

So we're given this information here that if we sell a DVD player how we sell 12 units and then if we sell it for 200 we sell 500 units. So we're going to, um, find a function of the demand as, ah, very boat or a function of the demand as a function of the price. Okay, so the demand here is the 12 or is how many units sold? So this is the demand, and we're going to call that, uh, variable d and then this is gonna be our price. But I'm gonna call variable p. So are the slope for our demand is going to be the change in demand in demand. Um, divided by the change in price. So are changing a man it's going to be equal to. So, for example, if we go from ah 15 or from 12 to 15 are change is going to be 15 minus 12 divided by and then ah, we went from 2 50 to 200 which means that we go 200 minus 250. So our slope then for our function, is when a vehicle to negative and then we have or sorry, three divided by negative 50. So this is the slope of our demand function. Now we need to plug it into a slope point slope form. So we're gonna have demand, uh, demand minus. Let's say some initial demand with Swiss say 12 or let's do 15 do U minus 15 and then this is gonna be good to our slope. So three or negative, 3/50 times. Let's say our initial price WAAS. So our price with the 15 units is 200. So we'll do p um minus 200. So then now we're gonna have that demand. It's gonna be equal to so negative 3/50 times p. And then this times this it's going to be able to will have plus, so 600 divided by 50 that's gonna be able to 12 and then we're gonna move this over here as well. So then it becomes plus 15. So our demand function is going to be negative. 3/50 p and then plus 27. So now we need to find the range of our, um, well, our domain. So if you notice here, um, if we increase in price for going to decrease in demand. So increase in price means a decrease in demand are an Until eventually that will get our demand is going to be equal to zero. So if we have a demand of zero, then our demand so negative 3/50 p plus 27 in order to reach a demand of zero than our price, it's gonna have to be will solve four p. So negative 27 is equal to negative 3/50 p, then times 50/3. So negative, 27 times negative, 50/3. So first we can divide 27 by three. So that's gonna become nine. The also the plus will cancel out. So we have nine times 50. So then we're going to get a price of 450. So when are prices 450? Our demand will be zero. Also, our price cannot be negative, right? So our price has to be some positive number. So our price it's going to be in the range of zero, not including zero, since we have to charge something off until our maximum price is going to be 450 because at that price. Our demand will be zero any higher than that. And then, um who won't have anything or our demand will be negative, which can't happen.

Is increasing at a rate of $5600 per week. The demand and caused functions for the product are given by P. Is equal to 6000 minus 25 X. And we're told that the cost function C. Is equal to 2400 X plus 5200. We are told to find the rate of change of sales with respect to time. When weekly sales are 44 X. Is equal to 44 units. Now. For this problem is a good thing that to right then what you do now. So we know from we know that the product is represented by X. And we know the profit is P. And we know the revenues are, so the first thing we know we know that D. P. D. T. Is 5600. Yeah. And we know that that the we know that the the man function which is the product that is represented by P. Is equal to 6000 minus 25 X. And we know the cost function C. It's 2400 X plus 5200. Uh huh. Yeah. Mhm. And the weekly cells X. It's 44 units right now. We also know that the relationship between are and see the revenue and the cost. We know the total product is going to be revenue BNP Being the product being the revenue minus two cost and the the revenue are was equal to the product P times a weekly sales acts. We could call this equation one, equation two. Now if you substitute equation to an equation one together we could have P is equal to P X minus C. Thanks. Yeah, concerning the values for P. X. N. C. We have P. In fact they're not the X. We get P 6000 minus 25 X. Yeah. Okay minus the cost function. 2400 X. Was 5200. Oh using the district distribute of property. We have 6000 x minus 25 X squared minus 2400 x. Yeah minus 5200. We're getting negative 25 plus 3600 x. Now we have two differential. Take the differential of P respect to X. So we have the differential P. D. P. D. X. Dp DT. We know that becomes the derivative of the 25 X squared becomes negative 50 X plus 3600 dx DT. Yeah. Yeah. Okay. Well we know that our D. P. D. C. Is 5600. Now you could you could you could bring the now we want to isolate dX DT so you can multi put there by both sides by negative 50 X plus 3600. Then your dx D. T becomes 5600 provided by 3600 minus 50 X. We know from the problem, given that our X. Is equal to 44 you could substitute 44 4 X. And you could it you can put that into your calculator were left. The value for dX DT becomes four. Now, what this tells us is that the rate of change of the of sales with respect to time when the weekly sales is X is equal to 44. Your left foot essentially four units per week. So that's the rate of change. That's your man.

All right. So here we have a problem. Coming is increasing production at a rate of twenty five units per day. So if we let Q B the number of units produced, we're given the cue. Tea is twenty five units per day, and now we're told that the price as a function of units produced Q is linear. So it looks something like a cube plus B. Where is going to be slope of this line and be a slight intercept? And we're given these two points. So when Q zero p is equal to sixty and when Q is one hundred, he's seventy. Correct? Instead of backwards, close it at a price of seventy. The demand is okay, so actually that were switched when the demand is seventy after the price. So when the price is seventy, the man to zero, and then when the price is sixteen, the demand is one hundred. Okay, so the number and, uh, there's a number of units we should produces one hundred on the price of sixty. Okay, so we just need to find the same being. Well, that's just gonna be like the slow. Why to my erstwhile on ever extreme sex line seventy minus sixty over zero, one hundred Just negative intent. And B is the y intercept. But you can see the weiners after seventy. So we have p his ninety one ten que plus seventy. Okay. And so let's see now we want the rd t. So what is our house? Our plane and we are is the munis produced times the price per unit. So if we just plug in what we had to content Q seventy, it's probably best to go ahead and distribute the cue because I have to differentiate. Okay, so then grd t which is what we're looking for. The rate of change of revenue S o we get. We want to find a drd t the rate of change of revenue when Q is twenty, twice the day of production is twenty. So the CT is negative. One cute tio tio plus seventy q t Now it could just playing in everything we know crudity is negative. One fifth time's here's twenty t k. D. T is twenty five, seventy times five and the city So this is negative. One hundred a B c. D. You're seventeen. Fifty here soon and we get it one thousand six hundred fifty dollars per day, so I revenues increasing by one thousand six hundred fifty dollars per day.


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