A cone has a radius of $24\ \text{cm}$ and a total surface area of $1536\pi\ \text{cm}^2$.

1. Use the total surface area of a cone to find the slant height.

For a cone,
$\text{total surface area}=\pi r^2+\pi rl.$
Given $r=24$ and total surface area $=1536\pi$,
$\pi(24)^2+\pi(24)l=1536\pi$
$576\pi+24\pi l=1536\pi$
$24l=960$
$l=40\ \text{cm}.$

2. Find the perpendicular height of the cone using the radius and slant height.

The radius, perpendicular height and slant height form a right-angled triangle.
$h^2+24^2=40^2$
$h^2=1600-576=1024$
$h=32\ \text{cm}.$

3. Calculate the volume of the cone and show it matches the stated result.

For a cone,
$V=\frac{1}{3}\pi r^2h.$
$V=\frac{1}{3}\pi(24)^2(32)$
$V=\frac{1}{3}\pi(576)(32)$
$V=6144\pi\ \text{cm}^3.$

1. Equate the volume of the sphere to the volume of the cone from part (a).

Since the cone is recast into a solid sphere, the volume remains the same.

$\frac{4}{3}\pi r^3 = 6144\pi$

2. Solve for the radius of the sphere.

$r^3 = 6144 \times \frac{3}{4} = 4608$

$r = \sqrt[3]{4608} = 16.650\ldots$

So, to 3 significant figures,

$r = 16.7$

1. Use the total surface area formula for a sphere.

For a sphere,
$\text{Total surface area}=4\pi r^2.$

2. Substitute the sphere radius from part (b)(i) and calculate the surface area.

From part (b)(i), using the unrounded value
$r=\sqrt[3]{4608}.$
Hence,
$\text{Total surface area}=4\pi(\sqrt[3]{4608})^2$
$=3480.09\ldots$
$\approx 3480\ \text{cm}^2.$