Q29. - Algebra 1

Question

The table shows the population density for Nevada for various years.

 

Year	Years since	People/ square mile
1930	0	$\mathbf{0}\mathbf{.}\mathbf{8}$
1960	30	$\mathbf{2}\mathbf{.}\mathbf{6}$
1980	50	$\mathbf{7}\mathbf{.}\mathbf{3}$
1990	60	$\mathbf{10}\mathbf{.}\mathbf{9}$
2000	70	$\mathbf{18}\mathbf{.}\mathbf{2}$

 

a. The population density $\mathit{d}$ of a Nevada from 1930 to 2000 can be modeled by $\mathit{d}\mathbf{=}\mathbf{0}\mathbf{.005}{\mathit{y}}^{\mathbf{2}}\mathbf{-}\mathbf{0}\mathbf{.127}\mathit{y}\mathbf{+}\mathbf{1}$, where $\mathit{y}$ represents the number of years since . Identify the type of polynomial $\mathbf{0}\mathbf{.005}{\mathit{y}}^{\mathbf{2}}\mathbf{-}\mathbf{0}\mathbf{.127}\mathit{y}\mathbf{+}\mathbf{1}$.

b. What is the degree of the polynomial?

c. Predict the population density of Nevada 2000. Explain your method.

d. Predict the population density of Nevada 2030. Explain your method.

of Nevada  . Explain your method.

Step-by-Step Solution

Answer

a. The type of polynomial $0.005y^2-0.127y+1$ is quadratic.

b. The degree of the polynomial is 2.

c. The population density of Nevada 2020 is $\mathbf{30}\mathbf{.07}$ people/square mile.

d. The population density of Nevada 2030 is $\mathbf{38}\mathbf{.3}$ people/square mile.

Part a Step 1. Write the given polynomial.

a) The given polynomial is $0.005y^2-0.127y+1$.

b) The degree of the polynomial is the greatest degree of any term in           the polynomial.

c) The formula to predict the population density $d$ of a Nevada is:

     $d=0.005y^2-0.127y+1$

     Where $y$ represents the number of years since 1930

d) The formula to predict the population density $d$ of a Nevada is

     $d=0.005y^2-0.127y+1$

     Where $y$ represents the number of years since 1930.

Part a Step 2. Formula used.

$d=a{x}^2+bx+c$

Part a Step3. Calculation

a) The type of polynomial $0.005y^2-0.127y+1$ is quadratic because the greatest degree of the terms in the given polynomial is 2.

 

b) The given polynomial is $0.005y^2-0.127y+1$. From the given polynomial, it can be concluded that the greatest    degree is 2. Therefore, the degree of the polynomial is 2.

 c) $y$ is the number of years since 1930.The value of $y$ for 2020 is given by :

$\begin{array}{l}y=2020-1930\\ =90\end{array}$

Therefore, the value of $y$ for 2020 is 90. Substitute 90 for $y$ in the equation $d=0.005y^2-0.127y+1$ to predict the population density of Nevada for 2020. Therefore, it is obtained that:

$\begin{array}{l}d=0.005y^2-0.127y+1\\ =0.005{\left(90\right)}^2-0.127\left(90\right)+1\\ =0.005\left(8100\right)-0.127\left(90\right)+1\\ =40.5-11.43+1\\ =30.07\end{array}$

Therefore, the population density of Nevada 2020 is$30.07$ people/square mile.

d) $y$ is the number of years since 1930. The value of $y$ for2030 is given by:

$\begin{array}{c}y=2030-1930\\ =100\end{array}$

Therefore, the value of $y$ for 2030 is 100. Substitute 100 for $y$ in the equation $d=0.005y^2-0.127y+1$ to predict the population density of Nevada for 2030.

Therefore, it is obtained that:

$\begin{array}{l}d=0.005y^2-0.127y+1\\ =0.005{\left(100\right)}^2-0.127\left(100\right)+1\\ =0.005\left(10000\right)-0.127\left(100\right)+1\\ =50-12.7+1\\ =38.3\end{array}$

Therefore, the population density of Nevada 2030 is$38.3$ people/square mile.