1. **State the problem:** You are managing a 90-unit apartment building. The base rent is 20000 per unit with full occupancy. For every 1000 increase in rent, one unit becomes vacant. We want to find the rent hike $x$ that maximizes total revenue. 2. **Complete the table:** Given $x$ is the number of 1000-rent hikes, - Rent per apartment: $20000 + 1000x$ - Number of rentals: $90 - x$ - Revenue: $R(x) = (20000 + 1000x)(90 - x)$ The table values are: | $x$ | Rent per Apartment | Units Rented | Revenue $R(x)$ | |---|--------------------|--------------|----------------| | 0 | 20000 | 90 | $20000 \times 90 = 1800000$ | | 1 | 21000 | 89 | $21000 \times 89 = 1869000$ | | 2 | 22000 | 88 | $22000 \times 88 = 1936000$ | | 3 | 23000 | 87 | $23000 \times 87 = 1991000$ | | 4 | 24000 | 86 | $24000 \times 86 = 2064000$ | | 5 | 25000 | 85 | $25000 \times 85 = 2125000$ | | 6 | 26000 | 84 | $26000 \times 84 = 2184000$ | | 7 | 27000 | 83 | $27000 \times 83 = 2241000$ | | 8 | 28000 | 82 | $28000 \times 82 = 2296000$ | 3. **Set up the quadratic function:** $$R(x) = (20000 + 1000x)(90 - x)$$ Expanding: $$R(x) = 20000 \times 90 - 20000x + 1000x \times 90 - 1000x^2 = 1,800,000 + 90,000x - 20,000x - 1000x^2$$ Simplify: $$R(x) = -1000x^2 + 70000x + 1,800,000$$ 4. **Express in vertex form:** Complete the square: $$R(x) = -1000x^2 + 70000x + 1,800,000$$ Factor out $-1000$ from the first two terms: $$R(x) = -1000(x^2 - 70x) + 1,800,000$$ Complete the square inside the parentheses: Take half of 70: $\frac{70}{2} = 35$, square it: $35^2 = 1225$ Add and subtract 1225 inside: $$R(x) = -1000(x^2 - 70x + 1225 - 1225) + 1,800,000$$ $$= -1000((x - 35)^2 - 1225) + 1,800,000$$ Distribute: $$= -1000(x - 35)^2 + 1000 \times 1225 + 1,800,000$$ $$= -1000(x - 35)^2 + 1,225,000 + 1,800,000$$ $$= -1000(x - 35)^2 + 3,025,000$$ So vertex form is: $$R(x) = -1000(x - 35)^2 + 3,025,000$$ 5. **Interpret the graph:** The graph is a downward-opening parabola with vertex at $x=35$, $R=3,025,000$. This means the maximum revenue is 3,025,000 when the rent hike is 35 (i.e., 35000 increase). 6. **Find maximum revenue and rent:** Maximum revenue is at $x=35$ hikes. Rent per apartment: $$20000 + 1000 \times 35 = 20000 + 35000 = 55000$$ Maximum revenue: $$R(35) = 3,025,000$$ 7. **Effect of 2 vacancies per 1000 hike:** If each 1000 hike causes 2 vacancies, then units rented become $90 - 2x$. Revenue function: $$R_2(x) = (20000 + 1000x)(90 - 2x) = 20000 \times 90 - 40000x + 90000x - 2000x^2 = -2000x^2 + 50000x + 1,800,000$$ This parabola opens downward more steeply (coefficient $-2000$ vs $-1000$), so maximum revenue occurs at a smaller $x$ and the maximum revenue is less than before. **Summary:** - The original maximum rent hike is 35 with rent 55000 and revenue 3,025,000. - Doubling vacancies per hike reduces max revenue and optimal hike.