The table shows the actual retail market share of passenger cards from Ford Motor Company as a percentage of the U.S. market.
Year Market Share
1975 23.6%
1980 17.2%
1985 18.8%
1990 20.0%
1995 20.7%
A mathematical model for this date is given by f(x) = 0.04x^2 - 0.8x + 22 where 0 corresponds to 1975. Complete the table below to find what the model predicts.
a. Complete the table.
x Market Share f(x)
0 23.6
5 17.2
10 18.8
15 20.0
20 20.7
b. Use values of the modeling function, f to estimate Ford's market share in 2000 and in 2005.
c. Write a brief description of Ford's market share from 1975 to 2005.
a. To complete the table, we need to substitute the values of x into the given mathematical model, f(x) = 0.04x^2 - 0.8x + 22.
For x = 0, we have f(0) = 0.04(0)^2 - 0.8(0) + 22 = 0 + 0 + 22 = 22. Therefore, the predicted market share for Ford in 1975 is 22%.
For x = 5, we have f(5) = 0.04(5)^2 - 0.8(5) + 22 = 0.04(25) - 4 + 22 = 1 - 4 + 22 = 19. Therefore, the model predicts a market share of 19% for Ford in 1980.
For x = 10, we have f(10) = 0.04(10)^2 - 0.8(10) + 22 = 0.04(100) - 8 + 22 = 4 - 8 + 22 = 18. Therefore, the predicted market share for Ford in 1985 is 18%.
For x = 15, we have f(15) = 0.04(15)^2 - 0.8(15) + 22 = 0.04(225) - 12 + 22 = 9 - 12 + 22 = 19. Therefore, the model predicts a market share of 19% for Ford in 1990.
For x = 20, we have f(20) = 0.04(20)^2 - 0.8(20) + 22 = 0.04(400) - 16 + 22 = 16 - 16 + 22 = 22. Therefore, the predicted market share for Ford in 1995 is 22%.
Therefore, the completed table is as follows:
x Market Share f(x)
0 23.6% 22
5 17.2% 19
10 18.8% 18
15 20.0% 19
20 20.7% 22
b. To estimate Ford's market share in 2000 and 2005, we need to find the corresponding values of x for those years.
For the year 2000, we can use the formula x = (year - 1975), so x = 2000 - 1975 = 25. Plugging this value into the model, we have f(25) = 0.04(25)^2 - 0.8(25) + 22 = 0.04(625) - 20 + 22 = 25 - 20 + 22 = 27. Therefore, the model predicts a market share of 27% for Ford in 2000.
For the year 2005, we can use the same formula to find x: x = 2005 - 1975 = 30. Plugging this value into the model, we have f(30) = 0.04(30)^2 - 0.8(30) + 22 = 0.04(900) - 24 + 22 = 36 - 24 + 22 = 34. Therefore, the model predicts a market share of 34% for Ford in 2005.
c. From 1975 to 2005, Ford's market share fluctuated according to the mathematical model provided. The model predicts that Ford's market share started at 22%, dropped to around 17% in 1980, gradually increased to 20% in 1990, and then reached its peak at 22% in 1995. After that, the market share declined based on the model's predictions.