1. **State the problem:** We need to find the cost of linoleum to cover a dining room described by the region bounded by the y-axis ($x=0$), x-axis ($y=0$), the line $y=25-5x$, and the vertical line $x=10$. 2. **Find the area of the region:** The area is under the line $y=25-5x$ from $x=0$ to $x=10$ above the x-axis. 3. **Set up the integral for the area:** $$\text{Area} = \int_0^{10} (25 - 5x) \, dx$$ 4. **Calculate the integral:** $$\int_0^{10} 25 \, dx = 25x \Big|_0^{10} = 25(10) - 25(0) = 250$$ $$\int_0^{10} 5x \, dx = \frac{5x^2}{2} \Big|_0^{10} = \frac{5(10)^2}{2} - 0 = \frac{5 \times 100}{2} = 250$$ 5. **Subtract to find the area:** $$\text{Area} = 250 - 250 = 0$$ 6. **Check the calculation:** The area cannot be zero; re-examine the integral calculation. 7. **Recalculate the integral correctly:** $$\int_0^{10} (25 - 5x) \, dx = \int_0^{10} 25 \, dx - \int_0^{10} 5x \, dx = 250 - 250 = 0$$ This suggests the line hits the x-axis at $x=5$ (since $25 - 5x=0 \Rightarrow x=5$), so from $x=5$ to $x=10$, the line is below the x-axis, so the region bounded is only from $x=0$ to $x=5$. 8. **Adjust the limits of integration to $0$ to $5$:** $$\text{Area} = \int_0^{5} (25 - 5x) \, dx$$ 9. **Calculate the area:** $$\int_0^{5} 25 \, dx = 25 \times 5 = 125$$ $$\int_0^{5} 5x \, dx = \frac{5 \times 5^2}{2} = \frac{5 \times 25}{2} = 62.5$$ 10. **Subtract to find the area:** $$\text{Area} = 125 - 62.5 = 62.5 \, m^2$$ 11. **Calculate the cost:** Cost per $m^2$ is 100, so total cost is: $$100 \times 62.5 = 6250$$ 12. **Choose the closest option:** Php 6 000 (option c) is closest to Php 6 250. **Final answer:** The cost of linoleum needed is approximately Php 6 000.