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Problem 69

Average Revenue A company sells two products. The demand functions of the products are given by

$q_{1}=300-2 p_{1} \quad$ and $\quad q_{2}=500-1.2 p_{2}$

where $q_{1}$ units of the first product are demanded at price $p_{1}$ and $q_{2}$ units of the second product are demanded at price $p_{2}$ . The total revenue will be given by

$R=q_{1} p_{1}+q_{2} p_{2}$

Find the average revenue if the price $p_{1}$ varies from $\$ 25$ to $\$ 50$ and the price $p_{2}$ varies from $\$ 50$ to $\$ 75 .$ (Hint: Refer to Exercises $61-64 . )$

Answer

$\$ 34,833$

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## Discussion

## Video Transcript

so, however, from here finding the average reason you given that we have a product key. One risotto, 300 minus toe p one way a P one is price is the unit price, which is there, which has the range of 25 50. And cute, too, is a second product, which is good, too. Hold it. 500 minus one pointed toe Pito. It's the way of people is the unity price off. Second product in the range is 50 to 75. So then they leaving you is given by first product times. The unity price P Warren Press sick and product thing was the unit price P two second unit price. As you see, this can't be good too. This earth is good, too. We have 300 things p one, three hundreds times p weren, then in minus two p. One is clear. Two people on the square been for us. Q. Two times Peter means that five hundreds. Pete too Fry, founder and p two. Then press is not presses in minus 1.2 Peter a scrape. Then mind us 1.2 p. Two. It's Cray. So they asked toe find the average treason. You You remember every way that the average revenue can write. Average a video as a uh this is our symbol. By our conversation, we can say that a R stands for average revenue. You know that the average very you for a mere four average value is one over is one over. A a witch is going to which is area anti Braun. Guard is time. See Intergroup. They're integral at Berlin. Rise are then therefore for eggs. Exile. Why deal for extent, Dick both Boyd. So for us to find ever driven you, which is average value. We need this A Because we have air for a is this one But it's not any eggs. And why isn't e p weren and be too and hard to find ever the the area since the area to bone her. So we have our bonus. You have very off p is here from 25 to 50 and from a 52 owes 75 for Peter. Let's try to sketch there. Dale, The area this area is that if we have, this is P P one. Uh, this is too boy axis Just gotta pee too and this case is not Why is P to access in pure nuts? Inside that toe hurt some ico units here we have red port here. One is from 25 minutes 250.25 years. It separates that 25. Is he this supposition? Because of course, we can't find the 25 unions. He So you need 25 from 01 and 2 25 and that terrible. So 50 fifth And here are already here of 75 01 toe at 75 five here and her. All right, start with 15. Sorry. Read her 50 here on 75 here 75 75 Here. So for P one is from 25 2 15 Slavery for Pete is that four people are is this range and north of four P too. Isn't the range of 50 Yeah, up to 75. So as you see, the intersection is here means that I want a Yeah a is in this here, which is this one. So this shows that I want a Is goto here is from 25 tools to 50 means that it is the area. Is he had This is this range The land off These is 50 minus 25. Here is the rand is 75 mile s 15 which is about 25. This is 0 to 25 minutes. That's the area is 25 times 25 which is good. Toe Bay is goingto 600 When if I seeks sounded at 25. So so far so cold we have is we have our function. We have a grounder is sort It is easier now to get our average or even you ivory retrieving You're gonna be good toe warren over area which is good toe hundreds 625 Remember that we have raise of 25 25 means that this is a square off our side's 25 25 square is 625 Then we inter berate from very off Key tool is a 50 and the final P world is 75 very of p weren rawboned is 25 and the emperor brooms is a 50 Then what is their function or function is 300 three hundreds uh, 300 the pure one mine US tool Kiwanis, Korea Uh, have a North most piss here in. Continue here, then impress 500 p. Tour mine US 1 25 not twins 1.2 p. Two a scrape, then Ditto DP one dp weren't and cross the prosthesis. Yeah, and would have the p one and they'd be two deep. You warn and dip a toe operation off pure And Peter, this Harvard for you is good too. Any grew for It's good to one over us six Sandage When you're five sounder than 25 any grow from 50 2 75 on DA So let's try to integrate with respect to be worn So this is good too if we integrate your respect to pure And now here we have 300 times pureness clear over to which is good toe 100 in the 50 if you want to spread, then minus two. He, uh, P Warner's grad intake off people on a square is a p wanted to you people run three. That's appeared one, 43 No peace three, but P World Party over three. Sorry. Minus tool over three p. One cube. Good. Then here are 500 p. One Pete too for us five 100 You are then appeared to because this is constant off everyone on this is for us. 1.2 for us, 1.2 p. One. Peter course here is independence of people. Were this court stands So we have NJ go off. This one is people one Then I put the brown Clarence off tweet off p one which sport to 25 better 15 and then 0.550. Um uh then interval rate with respect to be too dp to we're happy to Is the price off the king product? So far, so good. This is good too. We get here. We're, like over a 625. Yes, injury grew 50 from 52. 75 Then if we pour the vory v p here on, put 25 Here we put so 50 on DDE 25 15 25 You were good. We have 115 then times 50 squared is 215 2500. 1250 Didn't mind us or 25 spread Just go to minus 600 25. So I'm in a good position. Yeah, Then minus to aura told divides by three then here we have. We're going to put the very 50 25 here and pick you, which is Goto 50. P Cube would put 50 Cube, my ass. 25 cube. Here's to you. My last 25 cube. It used my ass here. There is most pissed. So means that they can arise here. Yeah, So our average reading you. Hey, uh, condom be good too. Let me repeat this world because there's no space here. I can't Even Your Grace, it's here. And then they do. Yeah, it's gonna be curious. So this is a warn over sorry or 360 hundreds. 25 on over 625. So, having table from 50 toe 75. So if we put the body off or P one here, this is good. Too. 115 under than 50 times. 27. 50. Scratch my nose. 25 screams. 50. Scared my last 25 square. Yes. And then we're gonna report very off. Be one horse. Here is this minus tool my ass to over three. Asked to grow from three times, he'll have 50 cube 50 cube, 50 cubed, minus 25 cube, 25 2.3. Yes, I have to be very sport. Our double 25 then Cuba Here, then press here. Are you here? Up? He would have calculated up to here on dhe prayers. 500 time Z 50 minus 25. A proponent of my last lower browned. This is very off p weren times p too. Then minus one point tool Titans 2015 minus 25 which is go to 25 Peter de nuit crawlers. This crypt square brackets then d y we can do. Then we can continue if intramural directing with respect toe. So that here we calculate there are you after? So this is good too. If Rick or even this constant. This is 50 square on my last 25 sprayer. So have to regret this. And we have took a great piece because this is no independent. Ah, there is no pure no p to This can't be you Goto or kill Benny called this constant a small a So with the integrated, this one would have a diamond sippy Because thymus Peter is not depend on did. This is constant Not depend on Peter. So very rabbinical toe warning or it's had tried to have enough space to use this piece with her. We have a one over three hundreds. 25 but not 300. But 60 100 25 this value, then time. See, we have eight times p too. Because we are going to integrate with respect to Pito. Not why? Sorry. P two. Yeah. We're going to injure grates with respect to Peter means that this is constant. We have to Yeah. Then press, remember that this is 25 15 minus 25. 25 is 500 time was 25 soar over too. Because this is a peace. Peace. Andrew Grove P two is P two square over to So we're creating this one. This is go toe to 50. You're 15 2 50 times e 25 Peter Peter screen. Remember that Tear off Pete tours with respect to pee too is Peter's square Over to then a 500 off two is 52 is 250 59 hours 25 This is 25. This is correct Then minus I think he overheard of peace. Creer Yes, we have peace. Chri Yes. You have peace Clear here so King would have peace clear over a city piece. Gray of precedent is that it's pee. Too poor to press. One is P Cube. Pito Cube. This is pee Tube. Event is, um if you're SP too. Yes, Peter Cube. Peter Cube. Over three. This is good to throw 24 because the one point toe Dorothy Time Z 25 he took you. Then we put for your 75 and 25 not 50. This is good too. Hundreds, not 100 1 of us 625. Remember that we have so current rate this one A by no program and corner creatives. But here we heard a time Z 75. My last 50 here means that's his 25 a 25 a. Your car gets very off, eh? There were about death. So here we have 75 minus 75 square minus 50 scream. So this sorbet good through here? 50 we have. Okay. Well, do they card? Question. We've We've come together after him. Right? 50 time was 25 Ben time Z 75 square minus 25. Screen don't mind us. 0.4 times 25. 25 times for his 100 seances. 10 minus 10. Then we have 2075 cube. So when five cube miners 50 cube our care Now we have ever think what we are going to do is to use the curricula. Thank you. Create this for you. This one. This one on Get the answer. Jamie was premios carrick it and find this solution. And it would have to do some simplification bringing 625 so Oh, ah, our average or even you. Our eight are forgeries in you forever. Egidio breathing you? It's good to yeah. 150 50 square, minus 25 Got cancer. You see him? I give something here and in here and left tackle creating for those We get 10 calls and and 500 over three. God knows you're Greta and you get this is good too. Three. Yeah, A three, three point three three three. Those are Charles and doors. So this is the answer. This is the average oration. You? Our average revenue is 34. Charles seemed HR under that through three

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