Let's solve the problem step-by-step.
### Part (a): Finding the diameter of the new sphere
1. Determine the radii of the existing spheres:
- The radius [tex]\( r \)[/tex] is half of the diameter [tex]\( d \)[/tex].
- Diameter [tex]\( d_1 = 1 \)[/tex] cm, so radius [tex]\( r_1 = \frac{d_1}{2} = \frac{1}{2} = 0.5 \)[/tex] cm.
- Diameter [tex]\( d_2 = 6 \)[/tex] cm, so radius [tex]\( r_2 = \frac{d_2}{2} = \frac{6}{2} = 3 \)[/tex] cm.
- Diameter [tex]\( d_3 = 8 \)[/tex] cm, so radius [tex]\( r_3 = \frac{d_3}{2} = \frac{8}{2} = 4 \)[/tex] cm.
2. Calculate the volumes of the three spheres:
- The volume [tex]\( V \)[/tex] of a sphere is given by the formula [tex]\( V = \frac{4}{3} \pi r^3 \)[/tex].
- Volume of the first sphere: [tex]\( V_1 = \frac{4}{3} \pi r_1^3 = \frac{4}{3} \pi (0.5)^3 \)[/tex].
- Volume of the second sphere: [tex]\( V_2 = \frac{4}{3} \pi r_2^3 = \frac{4}{3} \pi (3)^3 \)[/tex].
- Volume of the third sphere: [tex]\( V_3 = \frac{4}{3} \pi r_3^3 = \frac{4}{3} \pi (4)^3 \)[/tex].
3. Calculate the total volume of the new sphere:
- Total volume [tex]\( V_{\text{total}} = V_1 + V_2 + V_3 \)[/tex].
4. Find the radius of the new sphere from its volume:
- The volume of the new sphere is [tex]\( V_{\text{total}} = \frac{4}{3} \pi R^3 \)[/tex] where [tex]\( R \)[/tex] is the radius of the new sphere.
- Solve for [tex]\( R \)[/tex]: [tex]\( R = \left( \frac{3 V_{\text{total}}}{4 \pi} \right)^{\frac{1}{3}} \)[/tex].
5. Find the diameter of the new sphere:
- Diameter [tex]\( D = 2R \)[/tex].
By following these calculations, we find that the diameter of the new sphere is approximately [tex]\( 9 \)[/tex] cm.
### Part (b): Finding the surface area of the new sphere
1. Use the radius [tex]\( R \)[/tex] found in part (a): The radius of the new sphere is approximately [tex]\( 4.5 \)[/tex] cm.
2. Calculate the surface area of the new sphere:
- The surface area [tex]\( A \)[/tex] of a sphere is given by [tex]\( A = 4 \pi R^2 \)[/tex].
- Substitute [tex]\( R \approx 4.5 \)[/tex] cm into the formula to find [tex]\( A \)[/tex].
By performing these steps, we find that the surface area of the new sphere is approximately [tex]\( 254.57 \)[/tex] cm².
So the answers are:
- The diameter of the new sphere is [tex]\( 9 \)[/tex] cm.
- The surface area of the new sphere is approximately [tex]\( 254.57 \)[/tex] cm².
