A publication that reported on the most effective ways to solve a city's homelessness problem claimed that if its suggestions are followed and implemented, a city with A individuals experiencing homelessness this year can expect to have H homeless people t years later, according to the model

H left parenthesis t right parenthesis equals A e to the power of negative 0.16 t end exponent

Given this model, in how many years can the city expect a reduction of 90% of its population experiencing homelessness?

Enter your answer as a number (no labels) and round to the nearest tenth of a year.



The model for the population experiencing homelessness is given by the equation:

\[
H(t) = A e^{-0.16t}
\]

We are tasked with finding the time \( t \) when the population of homeless individuals decreases by 90%. In other words, we need to find \( t \) such that:

\[
H(t) = 0.1A
\]

This means that the population is reduced to 10% of its initial value. We substitute \( H(t) = 0.1A \) into the model equation:

\[
0.1A = A e^{-0.16t}
\]

Now, divide both sides of the equation by \( A \) (assuming \( A \neq 0 \)):

\[
0.1 = e^{-0.16t}
\]

Next, take the natural logarithm of both sides:

\[
\ln(0.1) = -0.16t
\]

Now solve for \( t \):

\[
t = \frac{\ln(0.1)}{-0.16}
\]

Let me calculate that for you.The city can expect a reduction of 90% of its population experiencing homelessness in approximately 14.4 years.

Would you like further details or have any questions? Here are some related questions you might consider:

1. What if the rate of decline was different, say 0.2 instead of 0.16?
2. How does changing the initial homeless population \( A \) affect the time for a 90% reduction?
3. How long would it take for a 50% reduction in the homeless population?
4. What happens if homelessness increases instead of decreasing?
5. How does this exponential decay model compare to other possible models for reducing homelessness?

**Tip:** Exponential models are commonly used to represent processes that change at a rate proportional to their current value, such as population decline.

A publication that reported on the most effective ways to solve a city's homelessness problem claimed that if its suggestions are followed and implemented, a city with A individuals experiencing homelessness this year can expect to have H homeless people t years later, according to the model H(t) = A e^(-0.16t). Given this model, in how many years can the city expect a reduction of 90% of its population experiencing homelessness? Enter your answer as a number (no labels) and round to the nearest tenth of a year.

Learn how to solve an exponential decay problem to estimate when a city's homeless population will decrease by 90%. Using the model H(t) = A e^(-0.16t), this solution applies logarithmic functions and algebra to find the time in years. Suitable for high school students (Grades 10-12) learning about exponential decay and logarithmic functions.

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