Question

Kimmie sells custom T-shirts for $12 each at the flea market every Saturday She usually sells 36 T-shirts Kimmie surveys her customers about whether they would buy her T-shirts at a different price. She determines that for every $1 increase in price she would sell two fewer T-shirts and for even $1 decrease in price, she would sell two more T-shirts, Which quadratic function can Kimmie use to model price increments vs. total income so that she can find the price at which her income is maximized?​

Answer 1

Answer:

y =  -2*x² + 60*x

Step-by-step explanation:

The general form of a quadratic function is:

y = a*x² + b*x + c

We need to determine a  ;  b  ;  and c. For that we have three conditions therefore

Condition 1       at  x = 12 $ each T-shirts kimmie sells 36 units, then she gets

12* 36 = 432 $    or      y  =  432   and

432 = a(12)² + 12*b + c      or   432 = 144*a + 12*b + c

Second condition   selling at 13 $ each T-shirt  she sells 34 , then

13*34 = 442

442  = a(13)²  +  13*b  + c    or   442 = 169*a + 13*b + c

And the third condition

11*38  = 418

418 = a* ( 121)  + 11*b  + c         418  =  121*a  +  11*b  + c

We have a three equation system

432 =   144*a + 12*b + c         (1)

442  =   169*a + 13*b + c       (2)

418 =    121*a  +  11*b  + c       (3)

We need to solve it for a,  b  and  c

Subtracting (2) - (1)        10 = 25*a  +  b    and subtracting (2) - (3)

24 = 48*a  + 2*b

Then       b  =  10 -  25*a         and     24  =  48*a + 2*( 10 - 25*a )

24  =  48*a +  20  - 50*a

24 -  20  =   -2*a

4  =  - 2*a    

a  = - 2      and   b =  10 - 25*( -2)     b = 60

Finally  c   is:    432 =   144*a + 12*b + c  

432  =  144* ( -2) + 12*60  + c

432  = -  288  +  720  +  c

432  = 432  + c

c =  0

The quadratic function is:

y =  -2*x² + 60*x