1. **State the problem:** We are given the function $f(t) = 243 \left(\frac{3}{5}\right)^t$ which models the number of stray cats $t$ years after an animal control program started. 2. **Find $f(0)$ and interpret:** $$f(0) = 243 \left(\frac{3}{5}\right)^0 = 243 \times 1 = 243$$ This means at the start of the program ($t=0$), there were 243 stray cats in the town. 3. **Find $f(0.5)$ and interpret:** $$f(0.5) = 243 \left(\frac{3}{5}\right)^{0.5} = 243 \times \sqrt{\frac{3}{5}} \approx 243 \times 0.7746 = 188.2$$ This means about 188 stray cats are estimated half a year after the program started. 4. **Interpret the number 3 in the fraction $\frac{3}{5}$:** The number 3 in the numerator shows the decay factor's numerator, indicating the population decreases by a factor of $\frac{3}{5}$ each year, representing exponential decay. 5. **Graphing $f(t)$ for $0 \leq t \leq 4$:** A suitable window for $t$ is from 0 to 4. For $f(t)$, since $f(0)=243$ and it decreases, the $y$-axis can range from 0 to about 250 to see the decay clearly. --- **Summary for problem 3:** - $f(0) = 243$ (initial stray cats) - $f(0.5) \approx 188.2$ (cats after half a year) - The factor $\frac{3}{5}$ shows exponential decay. --- "slug": "stray cats", "subject": "algebra", "desmos": {"latex": "y=243 \left(\frac{3}{5}\right)^x","features": {"intercepts": true,"extrema": true}}, "q_count": 7 }