1 Answer

Kildareflare · Answer 1 · 2023-11-01T05:54:31+0000

We are told that the volumes between two spheres is 16/3, this means that:

Now the formula for the volume of a sphere is:

$V=(4\pi r^3)/(3)$

Replacing in the previous formula:

$((4\pi r^3_1)/(3))/((4\pi r^2_2)/(3))=(16)/(3)$

Simplifying:

Now we solve for the first radius which is the radius of the bigger sphere:

Now we determine the radius of the smallest ball using the formula for the surface area of a sphere:

$A_S=4\pi r^2_2$

Now we replace the value of the surface area:

$42.3=4\pi r^2_2$

Now we solve for the radius:

$(42.3)/(4\pi)=r^2_2$

Taking square root on both sides:

$\sqrt[]{(42.3)/(4\pi)}=r_2$

Solving the operation:

$1.83\operatorname{cm}=r_2$

Now we replace the value of the radius in the proportion:

$r^3_1=(16)/(3)(1.83\operatorname{cm})^3$

Solving the operations:

$r^3_1=32.7\operatorname{cm}^3$

Taking cubic root to both sides:

$r_1=\sqrt[3]{32.7\operatorname{cm}^3}$

Solving the operation:

$r_1=3.19\operatorname{cm}$

Now we use the formula for the surface area using the new radius:

$A_S=4\pi r^2_1$

Replacing the radius:

$A_S=4\pi(3.19cm)^2$

Solving the operation:

$A_S=127.9\operatorname{cm}^2$

Therefore, the surface area of the new ball is 127.9 square centimeters.

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