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Question

# SOCIAL SCIENCE: Diffusion of Information by Mass Media In a city of a million people, election results broadcast over radio and television will reach $N(t)=1,000,000\left(1-e^{-0.3 t}\right)$ people within t hours. Find when the news will have reached 500,000 people.

Solution

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The number of people who have heard the election results within $t$ hours is

$\color{#4257b2} 1000000(1-e^{-0,3t}).$

Let us calculate the number of hours it takes for $500000$ people to hear the news. We solve for $t$.

\begin{align*} 1000000(1-e^{-0,4t}) &= 500000 \\ 1-e^{-0,3t} &= \frac{500000}{1000000} \\ 1-e^{-0,3t} &= 0,5 \\ e^{-0,3t} &= 1-0,5 \\ e^{-0,3t} &= 0,5. \end{align*}

Since $t$ is in the exponent we take the natural log on both sides

\begin{align*} \ln(e^{-0,3t} ) &= \ln0,5 \\ -0,3t \cdot \ln{e} &=\ln0,5 \\ -0,3t &= \ln0,5 \\ t &= \frac{\ln0,5}{-0,3}=\frac{-0,69315}{-0,3} \\ \color{#c34632} t &\color{#c34632}= 2,31049 \approx 2,3. \end{align*}

Therefore, it would take approximately $\color{#c34632}2,3hours$ for $500000$ people to hear the election results.

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