Question

use the table feature of a graphing calculator to do this exercise. the value (in millions of dollars) of new orders for all manufacturing industries can be approximated by the function $h(x) = -5.32x^3 + 48.9x^2 + 20.6x + 4123$, where $x = 0$ corresponds to the year 2000. create a table that gives the value of the sales for the years 2002 - 2007. complete the table below | year | x | h(x) | | 2002 | □ | □ |

Explanation:

Step1: Determine x for 2002

Since \( x = 0 \) corresponds to 2000, for 2002, \( x=2002 - 2000=2 \).

Step2: Substitute x = 2 into h(x)

The function is \( h(x)=- 5.32x^{3}+48.9x^{2}+20.6x + 4123 \).
Substitute \( x = 2 \):
\[

$$\begin{align*} h(2)&=-5.32\times(2)^{3}+48.9\times(2)^{2}+20.6\times(2)+4123\\ &=-5.32\times8 + 48.9\times4+41.2 + 4123\\ &=-42.56+195.6 + 41.2+4123\\ &=( - 42.56+195.6)+41.2 + 4123\\ &=153.04+41.2+4123\\ &=194.24 + 4123\\ &=4317.24 \end{align*}$$

\]

For the years 2003 - 2007:

• 2003: \( x = 3 \)

\[

$$\begin{align*} h(3)&=-5.32\times27+48.9\times9 + 20.6\times3+4123\\ &=-143.64+440.1+61.8 + 4123\\ &=( - 143.64+440.1)+61.8+4123\\ &=296.46+61.8+4123\\ &=358.26+4123\\ &=4481.26 \end{align*}$$

\]

• 2004: \( x = 4 \)

\[

$$\begin{align*} h(4)&=-5.32\times64+48.9\times16+20.6\times4 + 4123\\ &=-340.48+782.4+82.4+4123\\ &=( - 340.48+782.4)+82.4+4123\\ &=441.92+82.4+4123\\ &=524.32+4123\\ &=4647.32 \end{align*}$$

\]

• 2005: \( x = 5 \)

\[

$$\begin{align*} h(5)&=-5.32\times125+48.9\times25+20.6\times5+4123\\ &=-665+1222.5+103+4123\\ &=( - 665+1222.5)+103+4123\\ &=557.5+103+4123\\ &=660.5+4123\\ &=4783.5 \end{align*}$$

\]

• 2006: \( x = 6 \)

\[

$$\begin{align*} h(6)&=-5.32\times216+48.9\times36+20.6\times6+4123\\ &=-1149.12+1760.4+123.6+4123\\ &=( - 1149.12+1760.4)+123.6+4123\\ &=611.28+123.6+4123\\ &=734.88+4123\\ &=4857.88 \end{align*}$$

\]

• 2007: \( x = 7 \)

\[

$$\begin{align*} h(7)&=-5.32\times343+48.9\times49+20.6\times7+4123\\ &=-1824.76+2396.1+144.2+4123\\ &=( - 1824.76+2396.1)+144.2+4123\\ &=571.34+144.2+4123\\ &=715.54+4123\\ &=4838.54 \end{align*}$$

\]

Answer:

Year	x	h(x) (in millions of dollars)
2003	3	4481.26
2004	4	4647.32
2005	5	4783.5
2006	6	4857.88
2007	7	4838.54

For the 2002 row specifically:

• x: 2
• h(x): 4317.24