the sales of a new health magazine has been growing exponentially since 2005. the annual sales t years after…
the sales of a new health magazine has been growing exponentially since 2005. the annual sales t years after 2005 is represented by the function m(t).\nm(t)=100000e^{0.2t}\nhow many years after 2005 will the annual sales be 2,000,000?\ndrag a logarithm to the box and its value to the line to show the answer.\nt = \n\nor\nt = (roundest to the nearest tenth of a year)\n\nln(20÷0.2) ln(20×0.2) 0.2×ln(20) ln(20)÷0.2 0.3 0.6 1.4 2.0\nlog(20÷0.2) log(20×0.2) 0.2×log(20) log(20)÷0.2 4.6 6.5 12.5 15.0
Answer
Explanation:
Step1: Set up the equation
Set $M(t)=2000000$ in the function $M(t) = 100000e^{0.2t}$, so we have $2000000=100000e^{0.2t}$.
Step2: Simplify the equation
Divide both sides of the equation by 100000, getting $\frac{2000000}{100000}=e^{0.2t}$, which simplifies to $20 = e^{0.2t}$.
Step3: Take natural - logarithm of both sides
Using the property $\ln(e^x)=x$, taking the natural - logarithm of both sides of $20 = e^{0.2t}$ gives $\ln(20)=\ln(e^{0.2t})$, so $\ln(20) = 0.2t$.
Step4: Solve for $t$
Divide both sides of the equation $\ln(20)=0.2t$ by 0.2 to isolate $t$. We get $t=\frac{\ln(20)}{0.2}=\ln(20)\div0.2$.
Step5: Calculate the value of $t$
Using a calculator, $\ln(20)\approx2.9957$, and $\frac{\ln(20)}{0.2}=\frac{2.9957}{0.2}=14.9785\approx15.0$.
Answer:
$t=\ln(20)\div0.2$ $t = 15.0$