Answer:
The total surface area of the seven little spheres is 1.91 times the total surface area of the bigger sphere.
Explanation:
Volume of a Sphere
The volume of a sphere of radius r is given by:
[tex]\displaystyle V=\frac{4}{3}\cdot \pi\cdot r^3[/tex]
The volume of each little sphere is:
[tex]\displaystyle V_l=\frac{4}{3}\cdot \pi\cdot 2^3[/tex]
[tex]V_l=33.51\ mm^3[/tex]
When the seven little spheres coalesce, they form a single bigger sphere of volume:
[tex]V_b=7*V_l=234.57\ mm^3[/tex]
Knowing the volume, we can find the radius rb by solving the formula for r:
[tex]\displaystyle V_b=\frac{4}{3}\cdot \pi\cdot r_b^3[/tex]
Multiplying by 3:
[tex]3V_b=4\cdot \pi\cdot r_b^3[/tex]
Dividing by 4π:
[tex]\displaystyle \frac{3V_b}{4\cdot \pi}= r_b^3[/tex]
Taking the cubic root:
[tex]\displaystyle r_b=\sqrt[3]{\frac{3V_b}{4\cdot \pi}}[/tex]
Substituting:
[tex]\displaystyle r_b=\sqrt[3]{\frac{3*234.57}{4\cdot \pi}}[/tex]
[tex]r_b=3.83\ mm[/tex]
The surface area of the seven little spheres is:
[tex]A_l=7*(4\pi r^2)=7*(4\pi 2^2)=351.86\ mm^2[/tex]
The surface area of the bigger sphere is:
[tex]A_b=4\pi r_b^2=4\pi (3.83)^2=184.33\ mm^2[/tex]
The ratio between them is:
[tex]\displaystyle \frac{351.86\ mm^2}{184.33\ mm^2}=1.91[/tex]
The total surface area of the seven little spheres is 1.91 times the total surface area of the bigger sphere.
