Chapter 4.5, Problem 53E

(a)

To determine

To write: a polynomial equation to find the year when about $24014000000 of athletic equipment is sold.

Answer to Problem 53E

The polynomial equation is $5t^3-63t^2+70t+600=0$ .

Explanation of Solution

Given information:

During a 10 year period, the amount of athletic equipment $E$ sold domestically can be modeled by $E\left(t\right)=-20t^3+252t^2-280t+21614$ , where $t$ is in year.

Calculation:

Substituting $E\left(t\right)=24014000000$ , the equation becomes,

  $\begin{array}{c}-20t^3+252t^2-280t+21614=24014000000\\ -20t^3+252t^2-280t-2400=0\\ -4\left(5t^3-63t^2+70t+600\right)=0\\ 5t^3-63t^2+70t+600=0\end{array}$

Therefore, the polynomial equation is $5t^3-63t^2+70t+600=0$ .

(b)

To determine

To list: the possible whole number solutions of the equation in part (a).

Answer to Problem 53E

The possible whole number solutions are 1, 2, 3, 4, 5, 6, 7, 8, 10.

Explanation of Solution

Given information:

During a 10 year period, the amount of athletic equipment $E$ sold domestically can be modeled by $E\left(t\right)=-20t^3+252t^2-280t+21614$ , where $t$ is in year.

Consider the domain when making list of possible solutions.

Calculation:

The leading coefficient 5 and the constant term is 600. So, the possible whole number solutions are 1, 2, 3, 4, 5, 6, 7, 8, 10.

(c)

To determine

To find: when $24014000000 of athletic equipment is sold using synthetic division.

Answer to Problem 53E

24014000000 of athletic equipment is sold in 5 years and 10 years.

Explanation of Solution

Given information:

During a 10 year period, the amount of athletic equipment $E$ sold domestically can be modeled by $E\left(t\right)=-20t^3+252t^2-280t+21614$ , where $t$ is in year.

Calculation:

Test possible solutions using synthetic division until a solution is found.

For $x=5$ ,

  $\begin{array}{ccccc}5& 5& -63& 70& 600\\ & & 25& -190& -600\\ & 5& -38& -120& 0\end{array}$

For $x=10$

  $\begin{array}{ccccc}10& 5& -63& 70& 600\\ & & 50& -130& -600\\ & 5& -38& -600& 0\end{array}$

Therefore, 24014000000 of athletic equipment is sold in 5 years and 10 years.