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. Complete the table:
x Market Share f(x)
0 23.6 22.0
5 17.2
10 18.8
15 20.0
20 20.7
To complete the table, we need to substitute the x-values into the given function, f(x) = 0.04x^2 - 0.8x + 22.
For x = 0, f(0) = 0.04(0)^2 - 0.8(0) + 22 = 22.0.
Now, let's continue to fill in the empty cells:
x Market Share f(x)
0 23.6 22.0
5 17.2 19.0
10 18.8 19.4
15 20.0 19.6
20 20.7 19.6
b. To estimate Ford's market share in 2000 and 2005, we need to find f(x) for x = 2000 and x = 2005.
For x = 2000:
f(2000) = 0.04(2000)^2 - 0.8(2000) + 22
= 0.04(4,000,000) - 1600 + 22
= 160,000 - 1600 + 22
= 158,422
So, Ford's predicted market share in 2000 is approximately 158,422%.
For x = 2005:
f(2005) = 0.04(2005)^2 - 0.8(2005) + 22
= 0.04(4,020,025) - 1604 + 22
= 160,801 - 1604 + 22
= 159,219
Therefore, Ford's predicted market share in 2005 is approximately 159,219%.
c. From 1975 to 2005, Ford's market share fluctuated. It increased from 23.6% in 1975 to its highest point of 20.7% in 1995. After that, it gradually declined to an estimated market share of around 19.6% in 2005. Overall, Ford experienced some growth during this period but also faced competition and saw a slight decrease in its market share.
To complete the table, we need to substitute the values of x from the given years (0, 5, 10, 15, 20) into the mathematical model f(x) = 0.04x^2 - 0.8x + 22 and calculate the corresponding market share values.
a. Completing the table:
x Market Share f(x)
0 23.6 22
5 17.2 20.9
10 18.8 21.2
15 20.0 20.9
20 20.7 20.0
b. To estimate Ford's market share in 2000 and 2005, we need to find the values of x corresponding to those years and substitute them into the equation f(x).
To find x for the year 2000:
Given that 0 corresponds to 1975 and x is the number of years past 1975, we can calculate x for 2000 as:
2000 - 1975 = 25
Substituting x = 25 into the equation f(x):
f(25) = 0.04(25)^2 - 0.8(25) + 22
f(25) = 0.04(625) - 20 + 22
f(25) = 25 - 20 + 22
f(25) = 27
Therefore, the model predicts Ford's market share to be 27% in the year 2000.
To find x for the year 2005:
Given that 0 corresponds to 1975 and x is the number of years past 1975, we can calculate x for 2005 as:
2005 - 1975 = 30
Substituting x = 30 into the equation f(x):
f(30) = 0.04(30)^2 - 0.8(30) + 22
f(30) = 0.04(900) - 24 + 22
f(30) = 36 - 24 + 22
f(30) = 34
Therefore, the model predicts Ford's market share to be 34% in the year 2005.
c. From 1975 to 2005, Ford's market share fluctuated according to the given data and the mathematical model. It started at 23.6% in 1975, reached a low of 17.2% in 1980, and then gradually increased to its highest point of 20.7% in 1995. After 1995, it slightly decreased over the next ten years, with an estimated market share of 20% in 2005. Overall, Ford's market share remained relatively stable during this 30-year period, with some variation but no significant upward or downward trend.