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A food truck caters an event attended by 100 guests. Every guest orders one of two possible dishes: salad Or turkey plate_ The price of each meal decreases as more ...

Question

A food truck caters an event attended by 100 guests. Every guest orders one of two possible dishes: salad Or turkey plate_ The price of each meal decreases as more of that particular type are ordered: The price of a salad is S1O.00 minus S0.0S for each salad ordered_ The price of a turkey plate is S12.00 minus S0.02 multiplied by the square of the number of turkey plates ordered_ Guests pay for their meal only after everyone has placed their Order Using differentiation; find the maximum revenue

A food truck caters an event attended by 100 guests. Every guest orders one of two possible dishes: salad Or turkey plate_ The price of each meal decreases as more of that particular type are ordered: The price of a salad is S1O.00 minus S0.0S for each salad ordered_ The price of a turkey plate is S12.00 minus S0.02 multiplied by the square of the number of turkey plates ordered_ Guests pay for their meal only after everyone has placed their Order Using differentiation; find the maximum revenue for the food truck Remember that the number of meals is a positive integer: Round revenue to the nearest cent: maximum revenue: $



Answers

Assume that relative maximum and minimum values are absolute maximum and minimum values.
A concert promoter produces two kinds of souvenir shirt; one kind sells for $$ 18,$ and the other for $$ 25 . The total revenue from the sale of $x$ thousand shirts at $$ 18$ each and $y$ thousand at $$ 25 each is given by $$R(x, y)=18 x+25 y$$ The company determines that the total cost, in thousands of dollars, of producing $x$ thousand of the $$ 18 shirt and $y$ thousand of the $$ 25
shirt is given by $C(x, y)=4 x^{2}-6 x y+3 y^{2}+20 x+19 y-12$ How many of each type of shirt must be produced and sold in order to maximize profit?

Yeah. Given demanding question between price and quantities the equal toe minus off exabyte three plus 100. Now in first part we have to find the venue as functional X Now we know that it even will be called to price multiplied with one Did it So I worry when you will be equal toe my Nestle's XY squared by three plus on daytime's picks. Now in second part we have to find dominoes. Even you You know that even you cannot be negative. Our are will be greater than equal to zero So we can write my Nestle excess square by three less on their times X will be greater than equal to you know we can do that is a question that excess square by three is there stand equal toe on their times Next From here we can write X is less 10 equal to 300 Now we know that quantity cannot be negative. The X will be greater than equal toe Jiro No, In next part we have to find a venue for 100 units sold. So we can say that here we luv x will be equal to 100. It is given, So we have to find a venue for X equal 200 Revenge will be minus all I'm really squared, divided by three plus ready square On solving this expression, we will get a very venue Article toe dollar 600 intelligent 666 point 66 Now Moto fourth part in this part we have to find quantity for with the venue is maximized and we have to find maximum revaluate will to maximize the venue we will differentiate revenge on differentiating we will get br by D X is equal to minus both to exploit three plus 100 No, we will again difference here dysfunction. Then we will get the square Oughta born be exits grad which is goingto minus off to buy three which is less Kenjiro So from here you can see that we will get maximum from the city venue function to get maximum, We will put the are by D X is equal to zero So from here they will put Were you there by D X equal to zero That will be minus two x by three plus 100 from here There you off packs will be equal to 150 so we have to find maximum re venue. Bear quantity did 150 So very when you will be are equal toe my Nestle's 1 50 Holy square divided by three bless 100 might deployed with 1 50 on solving this Expressen we will get revenge you which is Dolev 7500 Now in last part, we have to find what price companies who charge for maximum re venue. That means price poor maximum re venue so price will be minus off One bite three multiplied with X for which even use maximum which is 1 50 So this will be 1 50 bless 100 So this will be called to minus or 50 plus 100. It is equal to $50. So this is our price.

I need to find a function that describes the revenue coming in from a charter flight, and then I want to maximize that revenue. Before I write the function, though, let's make sure we understand all of the variables X is. The number of unsold seats says there are 100 possible seats on this plane. 100 minus X tells me the number of flying passengers. Now, how much is the cost per ticket? That's the third piece that we need to make sure we know the cost per ticket is a flat $200 plus $4 per person for each unsold seat, so I'm gonna have to add four x two that baseline of $200. Okay, how much revenue is coming in for this flight with the money coming in? Is the number of passengers times the cost for each of their tickets? That's 100 minus X Times 200 plus four x. Let's get rid of those parentheses that gives me 20,000 plus 200 X minus four X squared. And I'm going to rewrite that so that all of my terms are in descending order That's minus four X squared plus 200 X plus 20,000. Yes, that is the function that shows the revenue for my flight. Now, if you look at the X squared term, you can see that has a coefficient of negative four and negative coefficient means this is a downward facing parabolas. So the vertex is the maximum value for my function. So at that the Vertex, the X, the number of unsold seats is going to maximize my revenue. So I want to find what that number is. What X Give me my maximum value of why To do that, we're going to complete the square every term with an ex. I'm going to keep together every term without an X. In this case, 20,000. I'm gonna push off to the side by itself for right now to complete the square, X square should have a coefficient of one. So I'm going to factor out a minus four from every term that has an accident. And again that 20,000 is just going to sit by itself for the moment. Now, to complete the square, my ex term has a coefficient of negative 50. I need to take half of the coefficient and square it. That's the number I'm gonna put back up into my function. So I'm going toe ad 625. Now, I can't just add a number to a function, so I'm gonna subtract 625 as well so that my overall ah, net result of doing all this to my function. I haven't changed a thing. The net result is exactly the same as what it waas. So the pieces I need to complete the square Our X squared minus 50 x plus 625. I would like to move the minus 625 outside of these parentheses. But don't forget, everything in the parentheses is being multiplied by negative four. So when I move it outside the parentheses, I'm really moving. Plus 2500 which all combined with a 20,000 that's already out there. So the final form of this function is going to be negative for times X minus 25 squared plus 22,500. That tells me where my ver Texas it is at the 0.25. 22,500. So what does this tell us? Well, if you remember our X variable is the number of unsold seats. So the X coordinate of my Vertex tells me that at that maximum point I will have 25 unsold seats. The Y value is the value of my function by function is marking how much revenue I have coming in. So this means my revenue is $22,500 and it's going to be a maximum. Okay, now let's graph that this equation looks like I have just about everything I need to graph it. I know where my Vertex is. I know this is a downward facing parabola, and I know what the equation is for my revenue, I would like, though toe have maybe two more points just to make sure that I'm mapping the width of this problem correctly. Since my Vertex is at 25 let's go a little bit on either side. Let's see what my revenue function would be for 20 unsold seats and for 30 unsold seats. Don't forget parabolas are symmetrical. Since I'm going the same number of units on others, either side of the Vertex, the value of the function of thes two points I chose will be identical. So let me plug in 20 into my revenue function. That gives me negative four times negative. Five squared plus truth 22,500. That gives me a value of 22,400. And that will be the same for both of these points. Now I can graph, but the addition of these two points here, this is what my graph looks like. You can see My Vertex is at 25. 22,500. The two points that we found it is a downward facing parabola. So this is my function.

You're demanding question between price and quantity. It's given that is equal toe minus off X by six plus 100. And in first part we have to find you a new ed Functional X. So we know that the venue is equal to price multiplied to it Quantity. So here price be multiplied with quantity. Sorry. When you will be minus off XY squared by six less 100 times Knicks Now In second part we have to find domain off the venue. We know that even you cannot be negative. So every venue are will be greater than equal to Jiro. But you can right minus off actually square by six plus 100 times X will be greater than equal toe Jiro. So from here we can write actually square by six is less than equal toe 100 x and from here actually be less 10 equal to 600. No, we know that one did He cannot be negative. X will be greater than equal toe judo. No entire part. We have to find the venue If 200 units are sold the venue for 2200 that will be minus hold 200 square the right by six plus 100 times 200 on solving this expression, we will get a very venue That will be Dola 13,000 333 point Basically now in fourth part we have to find quantity. It's maximize the venue. And what is the maximum revenge to maximize this re venue? Really? You'd the friend. She'll method own difference here. This br by DX will be equal to my Nestle's two x by six less 100. Now we will differentiate again. This equation then in the square be weighed by he exits Grant that will be equal to minus off one by three which is left Kenjiro From here we can say that this will lose less 10 0 From here we will get maximum now To get Maxima, we will put the outer by D X equals Kojiro. From here we will write Jiro equal to minus off two x by six plus under from here, will you next will be equal to 300 now for 300 quantity, we will get our maximum. But even the maximum re venue are will be equal to my Nestle 300 square Be right by six less 100 multiplied with 300 on solving this express and we will get it even read dollar 15 told you. But this is our maximum re venue now. In last part we have to find Christ. Would the company's all to maximize revenge? That means for maximums the venue What will be the price priced for maximum revenge will be minus ALS X by six you hear x will be equal to 300 be wide by six bless 100. This will be minus or 50 plus 100 that will be equal to $50.

Here we have the table of values which shows us the prophet information for the different orders of MP three players. And according to the table, it looks like the maximum profit would be around 3375. So that takes us through part A and then for part B. What we want to do is plot the points from the table and look to see if we have a function. So we have the point. 110 3135. We have won 23,240. We have 1 33,315 way have 1 43,360 we have 1 53,375 and we have 1 63,360 and we have 1 73,315 Okay, so I have a little extra blip in here, which I need to erase. So there we see all the points and it does look like a function because for every X value, there is only one y value. So yes, it's a function. Okay. For part C, we want to write the function and determine its domain. So based on the information from the problem, we know that for the 1st 100 units the profit would be $30 for the units ordered beyond 100 the profit would be $29.85. And that's because they're subtracting 15 cents profit her order. And so to get the total profit, we would add the profit for the 1st 100 good profit for the rest. Okay, the way I determined that the profit for the 1st 100 was $30 per MP three player is because they cost $60 to make, but they sell them $90 for $90 so 90 minus 60 is 30 so $30 per amply P three player times, 100 MP three players for the 1st 100 No, after that, they're making $29.85 for each one because they're taking off a whole 15 cents. And how many MP three players are they making this profit on? Well, you have to take the total number that they sold and subtract the 1st 100 from it. So there's our function now. If we wanted to simplify it. What we could Dio is multiply 30 and 100 we get 3000 and then we can distribute the 29.85 times acts so 29.85 x minus 29 85 times 100. And then after that, we could combine our like terms. So 3000 minus 22,985 is going to be 15. So 15 plus 29 85 times X would be the prophet equation for the domain. Since this profit equation was based on them selling more than 100 P three players, it's going to be X is greater than or equal to 100.


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