solve the problem using a graphing calculator. 5. a population in a town started with 1650 people living in…

solve the problem using a graphing calculator. 5. a population in a town started with 1650 people living in it. the population of the town after t years is represented by the function a(t)=1650e^(-0.09t). after how many years will town have 10 people left living in it? graph the function on a graphing calculator or at desmos.com and use the graph to make the prediction.

solve the problem using a graphing calculator. 5. a population in a town started with 1650 people living in it. the population of the town after t years is represented by the function a(t)=1650e^(-0.09t). after how many years will town have 10 people left living in it? graph the function on a graphing calculator or at desmos.com and use the graph to make the prediction.

Answer

Explanation:

Step1: Set up the equation

Set $A(t)=10$ in the function $A(t) = 1650e^{-0.09t}$, so we have $10 = 1650e^{-0.09t}$.

Step2: Isolate the exponential term

Divide both sides by 1650: $\frac{10}{1650}=e^{-0.09t}$, which simplifies to $\frac{1}{165}=e^{-0.09t}$.

Step3: Take the natural - logarithm of both sides

$\ln(\frac{1}{165})=\ln(e^{-0.09t})$. Since $\ln(e^{x}) = x$, we get $\ln(\frac{1}{165})=-0.09t$.

Step4: Solve for t

We know that $\ln(\frac{1}{165})=-\ln(165)$. So, $t=\frac{-\ln(\frac{1}{165})}{0.09}=\frac{\ln(165)}{0.09}$. Using a calculator, $\ln(165)\approx5.1188$, then $t=\frac{5.1188}{0.09}\approx56.876$.

Answer:

$t\approx56.88$ years