a The formula given in the question gives the sales in millions of dollars, so let's write the target amount the same way.
$24 014 000 000 is 24 014millions of dollars
Now we set the given expression to this amount and rearrange the resulting equation to find a polynomial equation.
b According to the Rational Root Theorem, all whole-number solutions of the equation in Part A are factors of the constant term -2400. Since the formula models the sale in a 10-year period, we are only interested in positive solutions not greater than 10. Let's list all such factors.
1,2,3,4,5,6,8,10
c We are asked to use synthetic division to find the integer solutions. Let's start with the smallest possible value, t=1.
rl IR-0.15cm r 1 & |rr -20& 252& -280& -2400 & -20& 232& -48 & c -20& 232 & -48 & -2448
The last entry in the last row is -2448, which is not 0, so t=1 is not a solution.
Let's move on to try t=2.
rl IR-0.15cm r 2 & |rr -20& 252& -280& -2400 & -40& 424& 288 & c -20& 212& 144& -2112
The last entry in the last row is -2112, which is not 0, so t=2 is not a solution.
Let's move on to try t=3.
rl IR-0.15cm r 3 & |rr -20& 252& -280& -2400 & -60& 576& 888 & c -20& 192 & 296 & -1512
The last entry in the last row is -1512, which is not 0, so t=3 is not a solution.
Let's move on to try t=4.
rl IR-0.15cm r 4 & |rr -20& 252& -280& -2400 & -80& 688& 1632 & c -20& 172 & 408 & -768
The last entry in the last row is -768, which is not 0, so t=4 is not a solution.
Let's move on to try t=5.
rl IR-0.15cm r 5 & |rr -20& 252& -280& -2400 & -100& 760& 2400 & c -20& 152 & 480 & 0 âś“
The last entry in the last row is 0, so t=5 is a solution.
Using the entries in the last line, we also get the factor form of the expression.
-20t^3+252t^2-280t-2400=(t-5)(-20t^2+152t+480)
To check whether the remaining values, 6, 8, and 10 are solutions, we substitute them in the quadratic factor.
t
-20t^2+152t+480
Value
6
-20( 6^2)+152( 6)+480
672
8
-20( 8^2)+152( 8)+480
416
10
-20( 10^2)+152( 10)+480
0 âś“
We found two whole-number solutions of the equation between 1 and 10.
t=5 and t=10
This means that there are two times, in the 5th year and in the 10th year, when $24 014 000 000 of athletic equipment is sold.
