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Problem 83 Easy Difficulty

A small theater has a seating capacity of 2000. When the ticket price is $ \$20 $, attendance is 1500. For each $ \$1 $ decrease in price, attendance increases by 100.

(a) Write the revenue $ R $ of the theater as a function of ticket price $ x $.
(b) What ticket price will yield a maximum revenue? What is the maximum revenue?

Answer

a) $R(x)=\left\{\begin{array}{ll}{2000 x} & {, x<15} \\ {3500 x-100 x^{2}} & {, 15 \leq x \leq 35}\end{array}\right.$
b) $17.50,30625$

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Video Transcript

this question consists off to parts We will start with party In this part, we have to write the venue are off the theater as a function off ticket price X. Now let us consider the function is it affects equals toe X square sbx Let's see. This will mark as equation number one. Now it is given that when the ticket prices $20 then the attendances 1500 to the venue is out of 20 questo 1500 multiplied by 20 out of twenties $30,000 this week will mark as equation number two. Also given in the caution is for each $1 decrease in ticket price. Attendance in theater increases by 100 and maximum capacity off theater is 2000, so the attendance for ticket price $19 is 1600 or ticket price, $18 a tendencies. 1700 all $17 attendances 1800 for $16. Attendance will be 1900 for $15. Attendance will be 2000, so the re venues will be out of 19 equals. Toe 1600 multiplied by 19. British $30,400 this week will mark as equation number three Similarly, the venue out of 15 was to 30,000 extender dollars. This will my education number four now by by using the equations 23 and four, we will deter mined the variables A, B and C So a multiplied by 20 square US be times 20 plus key was to 30,000. This will mark its equation number five eight times 19 square plus three times 19 plus Steve equals toe 30,400 This people markets equation number six a time stating squares plus three times 18 plus e equals two 30,600. This people markets Question number seven. Now on solving the equations 56 and seven we get a close to minus center equals 2 3500 and sees feel so the venue function is out of X equals toe minus 100 X square blows 3500 x You're X greater than equals two. 15 and less than equals to 20. Now they will do party in this part, we have to find out the ticket price which will the maximum re venue. The venue function is given by artifacts equals toe minus 100 Expert Yes, 3500 X and excess greater than equals to 15 and less than equals 20. These people Marcus Equation number mhm Now, from the question is, it is clear that it is less than you know. So the ticket price at which we will get the maximum revenge Aries X equals two minus B upon the way X equals two, putting the value of B and E buenos 4500 divided by minus two in 200. On dividing it, we will get the value of facts. The just $17.50 now revenge You had 17.5 year old dollars ability models 17.50 equals toe minus ender times 17.50 square plus 3500 times 17.50 mhm. In solving this, we will get value driven US 30,625 dollars. So the ticket price for maximum revenge you will be $17.50 and the value off maximum revenge you will be $30,625. Thank you

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