1. **Problem statement:** Two cubes each with volume 12.5 cm³ are joined end to end. We need to find the Lateral Surface Area (LSA), Total Surface Area (TSA), and volume of the resulting solid. 2. **Step 1: Find the side length of each cube.** The volume $V$ of a cube is given by: $$V = s^3$$ where $s$ is the side length. Given $V = 12.5$ cm³, solve for $s$: $$s = \sqrt[3]{12.5}$$ Calculate: $$s = 2.5 \text{ cm}$$ 3. **Step 2: Find the volume of the resulting solid.** Since two cubes are joined end to end, the total volume is: $$V_{total} = 2 \times 12.5 = 25 \text{ cm}^3$$ 4. **Step 3: Find the dimensions of the resulting solid.** Joining two cubes end to end along one edge creates a rectangular prism with dimensions: Length = $2s = 5$ cm Width = $s = 2.5$ cm Height = $s = 2.5$ cm 5. **Step 4: Calculate the Lateral Surface Area (LSA).** LSA of a rectangular prism is the sum of the areas of the four vertical faces: $$LSA = 2h(l + w)$$ Substitute values: $$LSA = 2 \times 2.5 \times (5 + 2.5) = 5 \times 7.5 = 37.5 \text{ cm}^2$$ 6. **Step 5: Calculate the Total Surface Area (TSA).** TSA is the sum of the areas of all six faces: $$TSA = 2(lw + lh + wh)$$ Calculate each term: $$lw = 5 \times 2.5 = 12.5$$ $$lh = 5 \times 2.5 = 12.5$$ $$wh = 2.5 \times 2.5 = 6.25$$ Sum: $$12.5 + 12.5 + 6.25 = 31.25$$ Multiply by 2: $$TSA = 2 \times 31.25 = 62.5 \text{ cm}^2$$ **Final answers:** - Volume = 25 cm³ - LSA = 37.5 cm² - TSA = 62.5 cm²