Given that from 1995 to 2003, the amount of mail M (in billions of pieces) handled by the U.S. Postal Service can be modeled by
[tex]M=0.05(t^4-18t^3+89t^2-32t+3680)[/tex]
where t is the number of years since 1995. In which year was there about 204,000,000,000 pieces of mail handled?
Part a:
Write a polynomial equation that can be used to answer the question.
The polynomial equation that can be used to answer the question is given by
[tex]0.05(t^4-18t^3+89t^2-32t+3680)=204 \\ \\ t^4-18t^3+89t^2-32t+3680= \frac{204}{0.05} \\ \\ t^4-18t^3+89t^2-32t+3680=4,080 \\ \\ t^4-18t^3+89t^2-32t-400=0[/tex]
Part b.
List the possible whole-number solutions of the equation in part (a) that are less than or equal to 8.
The possible whole-number solutions of the equation in part (a) are the factors of 400.
Therefore, the possible whole-number solutions of the equation in part (a) that are less than or equal to 8 are 1, 2, 4, 5, 8
Part c:
Use synthetic division to determine which of the possible solutions in part (b) is an actual solution. Then answer the question in the problem statement.
Substituting t = 1 into the polynomial from part a, we have
[tex](1)^4-18(1)^3+89(1)^2-32(1)-400=1-18+89-32-400 \\ \\ =-538\neq0[/tex]
Substituting t = 2 into the polynomial from part a, we have
[tex](2)^4-18(2)^3+89(2)^2-32(2)-400=16-18(8)+89(4)-64-400 \\ \\ =-448-144+356=-236\neq0[/tex]
Substituting t = 4 into the polynomial from part a, we have
[tex](4)^4-18(4)^3+89(4)^2-32(4)-400=256-18(64)+89(16)-128-400 \\ \\ =-272-1,152+1,424=0[/tex]
Therefore, the year that there was about 204,000,000,000 pieces of mail handled is in 1995 + 4 = 1999
