1. **State the problem:** Calculate the volume of a house-shaped figure composed of a rectangular prism base and a triangular prism roof. 2. **Given dimensions:** - Rectangular base: length $45$ cm, width $17$ cm, height $20$ cm - Triangular prism roof: slant height $18$ cm (height of the triangular face), base length $45$ cm, width $17$ cm 3. **Volume formulas:** - Volume of rectangular prism: $$V_{rect} = \text{length} \times \text{width} \times \text{height}$$ - Volume of triangular prism: $$V_{tri} = \frac{1}{2} \times \text{base} \times \text{height} \times \text{width}$$ 4. **Calculate volume of rectangular base:** $$V_{rect} = 45 \times 17 \times 20 = 15300$$ 5. **Calculate volume of triangular prism roof:** - Base of triangle = $45$ cm - Height of triangle = $18$ cm (slant height is the height of the triangular face) - Width (depth) = $17$ cm $$V_{tri} = \frac{1}{2} \times 45 \times 18 \times 17$$ Calculate step-by-step: $$\frac{1}{2} \times 45 = 22.5$$ $$22.5 \times 18 = 405$$ $$405 \times 17 = 6885$$ So, $$V_{tri} = 6885$$ 6. **Total volume:** $$V_{total} = V_{rect} + V_{tri} = 15300 + 6885 = 22185$$ 7. **Check your teacher's answer:** Your teacher's answer is $11645$, which is about half of $22185$. This suggests the slant height $18$ cm might not be the full height of the triangular face but the slant height of the roof. The actual height of the triangular face (perpendicular height) is needed. 8. **Find the perpendicular height of the triangular face:** - The triangular face is a right triangle with slant height $18$ cm and half the width $\frac{17}{2} = 8.5$ cm as base. Use Pythagoras theorem: $$h = \sqrt{18^2 - 8.5^2} = \sqrt{324 - 72.25} = \sqrt{251.75} \approx 15.87$$ 9. **Recalculate volume of triangular prism with perpendicular height:** $$V_{tri} = \frac{1}{2} \times 45 \times 15.87 \times 17$$ Calculate step-by-step: $$\frac{1}{2} \times 45 = 22.5$$ $$22.5 \times 15.87 \approx 357.075$$ $$357.075 \times 17 \approx 6060.275$$ 10. **Total volume with corrected height:** $$V_{total} = 15300 + 6060.275 = 21360.275$$ 11. **Teacher's answer is $11645$, which is about half of this.** Possibly the width used for the triangular prism is $8.5$ cm (half the base) instead of $17$ cm. 12. **Try volume of triangular prism with width $8.5$ cm:** $$V_{tri} = \frac{1}{2} \times 45 \times 15.87 \times 8.5$$ Calculate step-by-step: $$22.5 \times 15.87 = 357.075$$ $$357.075 \times 8.5 = 3035.14$$ 13. **Total volume:** $$V_{total} = 15300 + 3035.14 = 18335.14$$ Still not $11645$. Another possibility is the height of the rectangular prism is $20$ cm including the roof, so the base height is less. 14. **If base height is $10$ cm instead of $20$ cm:** $$V_{rect} = 45 \times 17 \times 10 = 7650$$ $$V_{tri} = 3035.14$$ $$V_{total} = 7650 + 3035.14 = 10685.14$$ Close to $11645$, so check exact measurements. **Summary:** - Use perpendicular height of triangular face, not slant height. - Use correct dimensions for base and roof. **Final formula:** $$V = 45 \times 17 \times 20 + \frac{1}{2} \times 45 \times h \times 17$$ where $$h = \sqrt{18^2 - \left(\frac{17}{2}\right)^2}$$ **Answer:** $$V \approx 15300 + 6060 = 21360 \text{ cm}^3$$ **Your mistake:** You added $45 \times 17 \times 18$ instead of using half the base times the perpendicular height for the triangular prism volume.