1. **Problem statement:** From one face of a wooden cube with side length 14 cm, hemispheres of diameter 1.4 cm are scooped out. We need to find the maximum number of hemispheres that can be scooped out and the total surface area of the remaining solid. 2. **Step 1: Calculate the number of hemispheres that fit on one face.** - The face is a square of side 14 cm. - Each hemisphere has diameter 1.4 cm, so radius $r = \frac{1.4}{2} = 0.7$ cm. - The hemispheres are arranged in a grid pattern without overlapping. - Number of hemispheres along one side = $\frac{14}{1.4} = 10$. - Total hemispheres on the face = $10 \times 10 = 100$. 3. **Step 2: Calculate the surface area of the original cube.** - Surface area of cube = $6 \times \text{side}^2 = 6 \times 14^2 = 6 \times 196 = 1176$ cm$^2$. 4. **Step 3: Calculate the surface area removed and added by scooping hemispheres.** - Each hemisphere removes a circular area on the cube face equal to the base area of the hemisphere: $\pi r^2 = \pi \times 0.7^2 = 0.49\pi$ cm$^2$. - Total area removed from the face = $100 \times 0.49\pi = 49\pi$ cm$^2$. - Each hemisphere adds a curved surface area equal to half the surface area of a sphere: Curved surface area of hemisphere = $2\pi r^2 = 2\pi \times 0.7^2 = 0.98\pi$ cm$^2$. - Total curved surface area added = $100 \times 0.98\pi = 98\pi$ cm$^2$. 5. **Step 4: Calculate the total surface area of the remaining solid.** - The original face area is reduced by the base areas of the hemispheres and increased by the curved surfaces of the hemispheres. - New surface area = Original surface area - area removed + curved surface area added $$ = 1176 - 49\pi + 98\pi = 1176 + 49\pi $$ - Using $\pi \approx 3.1416$: $$ 49 \times 3.1416 = 153.9384 $$ - So, $$ \text{Total surface area} \approx 1176 + 153.9384 = 1329.94 \text{ cm}^2 $$ **Final answers:** - Maximum number of hemispheres scooped out = $100$ - Total surface area of the remaining solid $\approx 1329.94$ cm$^2$