Determine the algebraic modeling The personnel costs in the city of Greenberg were $9,500,000 in 2009. In the recent past the personnel costs have increased at the rate of 4.2% annually. To model this, the city manager uses the function displaystyle{C}{left({t}right)}={9.5}{left({1}{042}right)}^{t}text{where} {C}{left({t}right)} is the annual personnel costs,in millions of dollars, t years past 2009 1) Write the equation that can be solved to find in what year the personnel costs will be double the 2009 personnel costs. 2) Then solve the equation numerically (use the Table feature of your calculator) or graphically and determine the year.

Determine the algebraic modeling The personnel costs in the city of Greenberg were $9,500,000 in 2009. In the recent past the personnel costs have increased at the rate of 4.2% annually. To model this, the city manager uses the function displaystyle{C}{left({t}right)}={9.5}{left({1}{042}right)}^{t}text{where} {C}{left({t}right)} is the annual personnel costs,in millions of dollars, t years past 2009 1) Write the equation that can be solved to find in what year the personnel costs will be double the 2009 personnel costs. 2) Then solve the equation numerically (use the Table feature of your calculator) or graphically and determine the year.

Question
Modeling
asked 2021-01-19
Determine the algebraic modeling The personnel costs in the city of Greenberg were $9,500,000 in 2009. In the recent past the personnel costs have increased at the rate of \(4.2\%\) annually.
To model this, the city manager uses the function \(\displaystyle{C}{\left({t}\right)}={9.5}{\left({1}{042}\right)}^{t}\text{where}\ {C}{\left({t}\right)}\) is the annual personnel costs,in millions of dollars, t years past 2009
1) Write the equation that can be solved to find in what year the personnel costs will be double the 2009 personnel costs.
2) Then solve the equation numerically (use the Table feature of your calculator) or graphically and determine the year.

Answers (1)

2021-01-20
Given,
Personnel cost \(\displaystyle=\${9},{500.000}\) in 2009
Rate of interest \(= 4.2\%\) (annually)
The annual cost function,\(\displaystyle{C}{\left({t}\right)}={9.5}{\left({1.042}\right)}^{t}\)
a)Write the equation that can be solved to find in what year personnel cost will be double the 2009 personnel costs?
b)Solve the equation numerically and determine the year.
Personnel cost will be double the 2009 personnel cost \(\displaystyle={2}\times\${9},{500},{000}= \displaystyle\${19},{000},{000}\)
So we get,
\(\displaystyle{A}={P}{\left({1}+{r}\right)}^{t}\)
\(\displaystyle{A}={9500000}{\left({1}+{0.042}\right)}^{t}\)
\(\displaystyle{19000000}={9500000}{\left({1.042}\right)}^{t}\)
\(\displaystyle{19}={9.5}{\left({1.042}\right)}^{t}\)
Find the value for t,
\(\displaystyle{199.5}={\left({1.042}\right)}^{t}\)
\(\displaystyle{2}={\left({1.042}\right)}^{t}\)
Apply logarithmic on both side we get,
\(\displaystyle \log{{2}}={ \log{{\left({1.042}\right)}}^{t}}\)
We know that, \(\displaystyle{ \log{{m}}^{n}=}{n} \log{{m}}\)
\(\displaystyle \log{{2}}={t} \log{{\left({1.042}\right)}}\)
\(\displaystyle{t}=\frac{{ \log{{2}}}}{{ \log{{\left({1.042}\right)}}}}\)
\(\displaystyle{t}=\frac{{{0.3010299}}}{{{0.0178677}}}\)
\(t = 16.8477\)
\(\displaystyle{t}\stackrel{\sim}{=}{17}\)
After 17 years we get the value of personnel cost is double in 2009.
So, \(\displaystyle{2009}+{17}={2026}\)
Therefore we get, \(\displaystyle{19}={9.5}{\left({1.042}\right)}^{t},\) Year is 2026.
0

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