Answer :
We start with the formula for the volume of a sphere:
[tex]$$
V = \frac{4}{3}\pi r^3.
$$[/tex]
Given that the volume is [tex]$100.5\text{ cm}^3$[/tex], we substitute into the formula:
[tex]$$
\frac{4}{3}\pi r^3 = 100.5.
$$[/tex]
Step 1. Solve for [tex]\( r^3 \)[/tex]:
Multiply both sides by [tex]$\frac{3}{4\pi}$[/tex] to isolate [tex]\( r^3 \)[/tex]:
[tex]$$
r^3 = \frac{3 \cdot 100.5}{4\pi}.
$$[/tex]
Calculate the numerator:
[tex]$$
3 \times 100.5 = 301.5,
$$[/tex]
so
[tex]$$
r^3 = \frac{301.5}{4\pi}.
$$[/tex]
Step 2. Evaluate the denominator:
Since
[tex]$$
4\pi \approx 12.566370614359172,
$$[/tex]
we have
[tex]$$
r^3 \approx \frac{301.5}{12.566370614359172} \approx 23.992607671103222.
$$[/tex]
Step 3. Take the cube root:
To solve for [tex]$r$[/tex], take the cube root of both sides:
[tex]$$
r = \sqrt[3]{23.992607671103222} \approx 2.884202955114811.
$$[/tex]
Step 4. Round the result:
Rounding [tex]$2.884202955114811$[/tex] to the nearest whole number gives:
[tex]$$
r \approx 3.
$$[/tex]
Thus, the radius is [tex]$\boxed{3\text{ cm}}$[/tex].
[tex]$$
V = \frac{4}{3}\pi r^3.
$$[/tex]
Given that the volume is [tex]$100.5\text{ cm}^3$[/tex], we substitute into the formula:
[tex]$$
\frac{4}{3}\pi r^3 = 100.5.
$$[/tex]
Step 1. Solve for [tex]\( r^3 \)[/tex]:
Multiply both sides by [tex]$\frac{3}{4\pi}$[/tex] to isolate [tex]\( r^3 \)[/tex]:
[tex]$$
r^3 = \frac{3 \cdot 100.5}{4\pi}.
$$[/tex]
Calculate the numerator:
[tex]$$
3 \times 100.5 = 301.5,
$$[/tex]
so
[tex]$$
r^3 = \frac{301.5}{4\pi}.
$$[/tex]
Step 2. Evaluate the denominator:
Since
[tex]$$
4\pi \approx 12.566370614359172,
$$[/tex]
we have
[tex]$$
r^3 \approx \frac{301.5}{12.566370614359172} \approx 23.992607671103222.
$$[/tex]
Step 3. Take the cube root:
To solve for [tex]$r$[/tex], take the cube root of both sides:
[tex]$$
r = \sqrt[3]{23.992607671103222} \approx 2.884202955114811.
$$[/tex]
Step 4. Round the result:
Rounding [tex]$2.884202955114811$[/tex] to the nearest whole number gives:
[tex]$$
r \approx 3.
$$[/tex]
Thus, the radius is [tex]$\boxed{3\text{ cm}}$[/tex].
