Question

the following table shows the value ( b ), in billions of dollars, of new construction put in place in the united states during year ( t )

\begin{array}{|c|c|} hline t = \text{year} & b = \text{value (billions of dollars)} \\ hline 2000 & 811.1 \\ 2003 & 891.5 \\ 2006 & 1167.6 \\ 2009 & 935.8 \\ hline end{array}

(a) make a table showing, for each of the 3-year periods, the average yearly rate of change in ( b ). (round your answers to two decimal places.)

\begin{array}{|c|c|c|c|} hline \text{period} & 2000 \text{ to } 2003 & 2003 \text{ to } 2006 & 2006 \text{ to } 2009 \\ hline \text{rate of change (in billion dollars per year)} & __ & & __ \\ hline end{array}

(b) explain in practical terms what ( b(2004) ) means.

• ( b(2004) ) is the year in which new construction spending totaled 2004 (in billions of dollars)
• ( b(2004) ) is the year in which new construction spending exceeded 2004 (in billions of dollars)
• ( b(2004) ) is the value (in billions of dollars) of new construction put in place in the united states in 2004
• ( b(2004) ) is the value (in billions of dollars) of new construction put in place in the united states before 2004

estimate ( b(2004) ). (round your answer to two decimal places.)

____ billion dollars

(c) over what period was the growth in value of new construction the greatest?

• before 2003
• 2000 to 2003
• 2003 to 2006
• option with blue dot, e.g., 2003 to 2006

Explanation:

Part (a)

To find the average yearly rate of change, we use the formula for the average rate of change: $\frac{\Delta R}{\Delta t}=\frac{R(t_2)-R(t_1)}{t_2 - t_1}$.

Step 1: 2000 to 2003

• $t_1 = 2000$, $R(t_1)=631.1$; $t_2 = 2003$, $R(t_2)=801.5$
• $\Delta t=2003 - 2000 = 3$
• Rate of change: $\frac{801.5 - 631.1}{3}=\frac{170.4}{3}= 56.80$

Step 2: 2003 to 2006

• $t_1 = 2003$, $R(t_1)=801.5$; $t_2 = 2006$, $R(t_2)=1167.6$
• $\Delta t=2006 - 2003 = 3$
• Rate of change: $\frac{1167.6 - 801.5}{3}=\frac{366.1}{3}\approx122.03$

Step 3: 2006 to 2009

• $t_1 = 2006$, $R(t_1)=1167.6$; $t_2 = 2009$, $R(t_2)=935.8$
• $\Delta t=2009 - 2006 = 3$
• Rate of change: $\frac{935.8 - 1167.6}{3}=\frac{- 231.8}{3}\approx - 77.27$

Part (b)

of $R(2004)$
The function $R(t)$ gives the value (in billions of dollars) of new construction put in place in the United States during year $t$. So $R(2004)$ is the value (in billions of dollars) of new construction put in place in the United States in 2004. So the correct option is "C. $R(2004)$ is the value (in billions of dollars) of new construction put in place in the United States in 2004".

Estimate $R(2004)$

We can use linear approximation between 2003 and 2006. The rate of change from 2003 to 2006 is approximately $122.03$ (from part (a)). From 2003 to 2004, $\Delta t = 1$.

• $R(2004)=R(2003)+ \text{rate of change} \times \Delta t$
• $R(2003) = 801.5$, rate of change $= 122.03$, $\Delta t = 1$
• $R(2004)=801.5+122.03\times1 = 923.53$ (we can also use the average rate from 2000 - 2003 and 2003 - 2006, but linear approximation between 2003 - 2006 is more accurate as 2004 is between 2003 and 2006).

Part (c)

We compare the rates of change:

• 2000 - 2003: $56.80$
• 2003 - 2006: $122.03$
• 2006 - 2009: $- 77.27$

The largest rate of change (in terms of growth, positive value) is from 2003 to 2006.

Final Answers

Part (a)

Period	Rate of change (in billion dollars per year)
2003 to 2006	$122.03$
2006 to 2009	$-77.27$

Part (b)

• Explanation: $R(2004)$ is the value (in billions of dollars) of new construction put in place in the United States in 2004.
• Estimate of $R(2004)$: $\boldsymbol{923.53}$ (or other reasonable approximation based on method, but using linear approximation between 2003 - 2006 gives this)

Part (c)

The growth in value of new construction was the greatest from $\boldsymbol{2003}$ to $\boldsymbol{2006}$.

Answer:

of $R(2004)$
The function $R(t)$ gives the value (in billions of dollars) of new construction put in place in the United States during year $t$. So $R(2004)$ is the value (in billions of dollars) of new construction put in place in the United States in 2004. So the correct option is "C. $R(2004)$ is the value (in billions of dollars) of new construction put in place in the United States in 2004".

Estimate $R(2004)$

We can use linear approximation between 2003 and 2006. The rate of change from 2003 to 2006 is approximately $122.03$ (from part (a)). From 2003 to 2004, $\Delta t = 1$.

• $R(2004)=R(2003)+ \text{rate of change} \times \Delta t$
• $R(2003) = 801.5$, rate of change $= 122.03$, $\Delta t = 1$
• $R(2004)=801.5+122.03\times1 = 923.53$ (we can also use the average rate from 2000 - 2003 and 2003 - 2006, but linear approximation between 2003 - 2006 is more accurate as 2004 is between 2003 and 2006).

Part (c)

We compare the rates of change:

• 2000 - 2003: $56.80$
• 2003 - 2006: $122.03$
• 2006 - 2009: $- 77.27$

The largest rate of change (in terms of growth, positive value) is from 2003 to 2006.

Final Answers

Part (a)

Period	Rate of change (in billion dollars per year)
2003 to 2006	$122.03$
2006 to 2009	$-77.27$

Part (b)

• Explanation: $R(2004)$ is the value (in billions of dollars) of new construction put in place in the United States in 2004.
• Estimate of $R(2004)$: $\boldsymbol{923.53}$ (or other reasonable approximation based on method, but using linear approximation between 2003 - 2006 gives this)

Part (c)

The growth in value of new construction was the greatest from $\boldsymbol{2003}$ to $\boldsymbol{2006}$.