Please help with questions 9 and 10 7 The following data tab

Please help with questions 9 and 10!

7) The following data table shows the monthly profit P, in dollars, at a monthly production level of N video games. Should profit be modeled by a linear function or by a quadratic function? Number N 200 250 300 350 400 Profit P 3100 4375 5500 6475 7300 8) Why? Calculate differences to support your answer to #7

Solution

7. Profit P, is a function of number of video games, N and it should be modelled as a quadratic function.

8. Profit function should be modelled as a quadratic function as for consequent increase in N by 50, the increase in profit decreases more and more i.e profit is increasing but at a decreasing rate. For example for an increase in N from 200 to 250, the profit increases by 1,175 units from 3100 to 4375 and for the next increase in N from 250 to 300, the profit increases by 1,125 from 4375 to 5500, for the next increase in N from 300 to 350, profit increases by 975 from 5500 to 6475. Hence, one can see that with the increase in N, the profit is increasing but the increase is decreasing for every unit increase in N.

9. The profit function could be written as aN^2 + bN + c where we need to find out the value of a, b and c. Substituting three points in this equation we can figure out the values of a, b and c

1. 3100 = 40000a + 200b + c

Or 31 = 400a + 2b + c

2. 5500 = 90000a + 300b + c

55 = 900a + 3b + c

3. 7300 = 160000a + 400b + c

73 = 1600a + 4b + c

Substituting c from eq 1 in eq 2 and 3 yields

24 = 500 a + b

42 = 1200 a + 2b

Solving the above two equations yields a = - 0.03 and b = 39. Substituting these values in eq 1 yields c = -35

Hence, the profit function is P = -0.03N^2 + 39 N - 35

10. In the above equation, putting P = 0 yields 0.03 N^2 -39N + 35 = 0

Solving this quadratic using the determinant formula, D = (-39)^2 - 4(0.03)(35) = 1516.8

Two roots of this equation are (39 +/- (1516.8)^1/2)/2(0.03) = (39+/- 38.94)/0.06 = 1299, 1 . Hence at N = 1 and at N = 1299 the profit level would be zero