Isobel is producing and selling cassette tapes of her rock b

Isobel is producing and selling cassette tapes of her rock band. When she had sold 10 tapes, her net profit was $6. When she had sold 20 tapes, however, her net profit had shrunk to $4 due to increased production expenses. But when she had sold 30 tapes, her net profit had rebounded to $10. (a) Give a quadratic model relating Isobel\'s net profit y to the number of tapes sold x. y = (b) Divide the profit function in part (a) by the number of tapes sold x to get a model relating average profit w per tape to the number of tapes sold. w = (c) How many tapes must she sell in order to make $2. 38 per tape in net profit? tapes

Solution

quadratice model: y = ax^2 +bx +c

Three points ( 10, 6) , ( 20, 4) and ( 30 , 10)

6 = 100a +10b +c -----(1)

4 = 400a + 20b +c ----(2)

10 = 900a +30b +c -----(3)

solve the three equations to get the value of a , b , c:

a = 1/25 ; b =-7/5 ; c = 16

a) So, y = x^2/25 -7x/5 +16

b) Avg profit per tape , w = ( x^2/25 -7x/5 +16)/x = x/25 -7/5 +16/x

c)   x/25 -7/5 +16/x = $2.38

( x^2 -35x +80) = 2.38x

x^2 -37.38x +80 =0

x = 18.69 +/- 16.41 = 2.28 , 35.1

So, x = 35 tapes