1. **State the problem:** Find the inverse function of the model $$n = \frac{848r}{1000} - 1525$$ where $n$ is the number of playgrounds built and $r$ is the amount of money raised. 2. **Formula and important rule:** To find the inverse function, swap $n$ and $r$ and solve for the new $r$. 3. **Find the inverse function:** Start with the original function: $$n = \frac{848r}{1000} - 1525$$ Swap $n$ and $r$: $$r = \frac{848n}{1000} - 1525$$ Now solve for $n$: $$r + 1525 = \frac{848n}{1000}$$ Multiply both sides by $\frac{1000}{848}$: $$n = \frac{1000}{848}(r + 1525)$$ Intermediate step showing cancellation: $$n = \frac{\cancel{1000}}{\cancel{848}}(r + 1525)$$ Simplify fraction: $$n = \frac{125}{106}(r + 1525)$$ So the inverse function is: $$n = \frac{125}{106}(r + 1525)$$ 4. **Interpretation:** The inverse function gives the number of playgrounds $n$ built for a given amount of money $r$ raised. 5. **Calculate money raised if $40$ million was spent on playgrounds:** Given $r = 40$ (million), find $n$: $$n = \frac{848 \times 40}{1000} - 1525 = \frac{33920}{1000} - 1525 = 33.92 - 1525 = -1491.08$$ Since negative playgrounds don't make sense, this suggests the model expects $r$ in thousands of dollars, so $40$ million = $40000$ thousands. Recalculate with $r=40000$: $$n = \frac{848 \times 40000}{1000} - 1525 = 33920 - 1525 = 32395$$ So, approximately 32,395 playgrounds were built. 6. **Find the inverse function again (part c):** We already found the inverse function: $$r = \frac{1000}{848}(n + 1525)$$ Simplify: $$r = \frac{125}{106}(n + 1525)$$ 7. **Find dollars raised if 39,179 playgrounds were built (part d):** Use inverse function: $$r = \frac{125}{106}(39179 + 1525) = \frac{125}{106} \times 40704$$ Calculate: $$r = 1.179245 \times 40704 \approx 47982.5$$ So, approximately 47,983 thousands of dollars, or 47.983 million dollars were raised. **Final answers:** - Inverse function: $$r = \frac{125}{106}(n + 1525)$$ - Money raised for 39,179 playgrounds: approximately 47,983 thousands dollars.