POTM+6

​Problem of the month (POTM) #6

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Movie Ticket Problem A movie theater charges $7.00 per ticket. At that price, theater owners can expect to sell 1,100 tickets. They also found that for every 10 cent increase in ticket price, they will sell 20 fewer tickets. However, for every 10 cent reduction in the ticket price, they will sell an additional 20 tickets. What ticket price will maximize their income and what will their maximum income be?

// Hints: // Student work: 1 -If we let y equal the income and then we make a table to but into a stat plot we might be able to graph this as a quadratic equation. 2 - Yea, that's a good idea! I was thinking about graphing it too. This is like what we're learningish! I like what amigo tres y cuatro said. Those are good ideas. Wait, I'm a little confused. #3: Why would the numbers in pink be the highest? Oh, nevermind, I got it. 3- I think we should graph it. if they charge $7, they earn $7700 per show $7.1, $7668 $6.5, $7800 $6.6, $7788 $6.4, $7808 $6.3, $7812  $6.25, $7812.5 HIGHEST BY 0.50 $6.2, $7812 $6.1, $7808 //$8, $7200// $6, $7800 //$5, $7500// 7>x>6__** Another thought: for every 1 cent change, they will increase or decrease 2 tickets. or every 1 dollar change, they will increase or decrease 200 tickets. let x=price y=x[1100+200(7-x)] I think that's the equation check: 7812=6.3[1100+200(7-6.3)] 7812=6.3[1100+140] 7812=7812 YAY I got it OK I graphed it on calculator and it said 7812.5 is the MAX. a estudiante tres, the pink one is the highest is because it's the maximum or something. 4 - If the tickets sell for $6.3, you will sell 1240 tickets, so the max. income will be $7812... but if you sell the tickets for $6.25, you are reducing the price by 5 cents, so you will sell 10 additional tickets. $6.25 * 1250 = $7812.50 5 -I think we should solve each seperately first and make a graph for something... Like use the formula and do the number of tickets sold times price per ticket when the price goes up and use the fomrula to figure out how the income goes when the price goes down from 7 dollars. Then, since there should be like a 'maximum' in the graph (double line graph?), because the incomes go up and down (generally), we just find the maximum point. And, that is the anwser. Right? 6 -I like what number 3 did by putting all the information down. I also agree that we should try and graph it to make it easier. Why are we speaking in spanish I don't know spanish. :( 7 -you wouldn't make a double line graph because if you look at the probem above it has two x's since it has two x's what you would get is y=2500x-200x^2 then you would find the maximun with is x=6.249993 and y= 7812.5 then you round the x to the nearest hundreth and you would get $6.25 :D 8 - OMG THIS IS SO CONFUSING! Never mind I understand now. I subtracted .1 from every ticket price and added 20 tickets every time starting from the original price. 9 - I agree with #s 3-5 saying how 7812 is the max number. 10 - I was a little confused at first but then after looking at what #3 wrote, i understood better and i agree that 7812 would be the max of the graph. 11 - First we have to find the original income which is 7700$. You get this by the formula for // Income = (number of tickets sold) x (price per ticket). // Then using this formula, you get two equations for the ten cent decrease and 20 ticket increase; and the ten cent increase, twenty ticket decrease. K, for the first one- you have to take it in two pieces- (number of tickets sold) and (price of ticket) 1100+20x---for number of tickets and 7-.10x---for price.... or i think right???? so it comes to (1100+20x) (7-.10x)= income for the 10 cent decrease. then you foil and then graph. You do the same thing with the ten cent increase and graph. After graphing both you can definitely see which one is right, (because one increases in salary while going DOWN and the other up sooooo yeaaah... you'll get it when you see....) Then you find the maximum and then substitute the number for x and thats how you find the answer... I was reaaaaaaaallllllllyyy confused when I got this problem, but then my friend explained this to me and I was like OHHHH!!!!!! so yeah, this is like not my original work because my friend helped me a LOOOOTTTTTT .......... kk? (just making that clear) :D 12 - 13 - 14 - 15 - 16 -
 * 1) // Income = (number of tickets sold) x (price per ticket) //
 * 2) // Use your calculator to assist your thoughts in solving this problem: //
 * __So they should charge between $6.3 to 6.2

** // 1 point // ** **State** what the problem is looking for. What ticket price will maximize their income and what will their maximum income be? ** // 2 points // ** ** Answer ** the problem correctly. The ticket price of $6.25 will maximize their income. The theater's maximum income will be $7,812.50. ** // 5 points // ** ** Explain the process ** you used to solve the problem.

** // 2 points // ** ** Justify **  your answer.