Respuesta :
Answer:
• The estimated minimum wage for 2013 is 8.55.
,• In 2017, the minimum wage will be around 10.00.
Explanation:
The general form of an exponential model is given as:
[tex]f(t)=ab^t[/tex]In 1970, The minimum wage was 1.60. i.e.
• When time, t = 0, f(t) = 1.60
[tex]\begin{gathered} 1.60=a(b)^0 \\ \implies a=1.60 \end{gathered}[/tex]Substitute a=1.60 into f(t).
[tex]f(t)=1.60(b)^t[/tex]Next, in 2000 it was 5.15.
• When t=2000-1970=30, f(t)=5.15.
Substitute these values into the formula above:
[tex]5.15=1.60(b^{30})[/tex]We solve the equation for b.
[tex]\begin{gathered} \text{ Divide both sides by 1.60} \\ \frac{5.15}{1.60}=b^{30} \\ 3.21875=b^{30} \\ \text{ Take the 30th root.} \\ b=3.21875^{\frac{1}{30}} \end{gathered}[/tex]Thus, our exponential model f(t) is:
[tex]f(t)=1.60(3.21875^{\frac{t}{30}})[/tex](a)Minimum Wage in 2013
In 2013, t=2013-1970=43
[tex]\begin{gathered} f(43)=1.60(3.21875^{\frac{43}{30}}) \\ f(43)=8.55 \end{gathered}[/tex]The estimated minimum wage for 2013 is 8.55.
(b)When the minimum wage, f(t)=10.00
We want to find the time, t.
[tex]\begin{gathered} f(t)=1.60(3.21875^{\frac{t}{30}}) \\ 10=1.60(3.21875^{\frac{t}{30}}) \end{gathered}[/tex]The equation is solved for t below:
[tex]\begin{gathered} \text{ Divide both sides by 1.60} \\ \frac{10}{1.60}=\frac{1.60(3.21875^{\frac{t}{30}})}{1.60} \\ 6.25=3.21875^{\frac{t}{30}} \\ \text{ Take the logarithm of both sides} \\ \log(6.25)=\log(3.21875^{\frac{t}{30}}) \\ \text{ By the power law of logarithms:} \\ \operatorname{\log}(6.25)=\frac{t}{30}\operatorname{\log}(3.21875^) \\ \text{ Multiply both sides by }\frac{30}{\operatorname{\log}(3.21875^)} \\ \operatorname{\log}(6.25)\times\frac{30}{\operatorname{\log}(3.21875)}=\frac{t}{30}\operatorname{\log}(3.21875)\times\frac{30}{\operatorname{\log}(3.21875)} \\ t=47.03 \end{gathered}[/tex]Thus, approximately 47 years after 1970, which is in 2017, the minimum wage will be 10.00.
