1. **State the problem:** We have a solid triangular prism with right-angled triangular bases. The base and height of the triangle are $20x$ cm and $15x$ cm respectively, and the hypotenuse is $25x$ cm. The length of the prism is $y$ cm. The total surface area is $3600$ cm². We need to find the relationship between $x$ and $y$ or solve for one variable. 2. **Formula for total surface area of a triangular prism:** $$\text{Surface Area} = 2 \times \text{Area of triangular base} + \text{Perimeter of base} \times \text{length}$$ 3. **Calculate the area of the triangular base:** $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 20x \times 15x = 150x^2$$ 4. **Calculate the perimeter of the triangular base:** $$\text{Perimeter} = 20x + 15x + 25x = 60x$$ 5. **Write the surface area equation:** $$3600 = 2 \times 150x^2 + 60x \times y$$ $$3600 = 300x^2 + 60xy$$ 6. **Isolate $y$:** $$3600 - 300x^2 = 60xy$$ $$y = \frac{3600 - 300x^2}{60x}$$ 7. **Simplify the fraction:** $$y = \frac{\cancel{60}(60 - 5x^2)}{\cancel{60} x} = \frac{60 - 5x^2}{x}$$ **Final answer:** $$y = \frac{60 - 5x^2}{x}$$ This expresses the length $y$ of the prism in terms of $x$ given the total surface area of 3600 cm².