To find the radius of a hemisphere with a volume of 1773 cm³, we can use the formula for the volume of a hemisphere, which is:
\(\[ V = \frac{2}{3} \pi r^3 \]\)
where \(\( V \)\) is the volume and \(\( r \)\) is the radius. Since we're given the volume, we need to solve for \(\( r \)\).
First, isolate \(\( r^3 \)\) by multiplying both sides by \(\( \frac{3}{2\pi} \)\) to cancel out the \(\( \frac{2}{3} \pi \)\) on the right side:
\(\[ r^3 = \frac{V \cdot \frac{3}{2}}{\pi} \]\)
Next, plug in the known volume of 1773 cm³:
\(\[ r^3 = \frac{1773 \cdot \frac{3}{2}}{\pi} \]\)
Now, calculate the numerical value inside the cube root:
\(\[ r^3 = \frac{1773 \cdot 1.5}{\pi} \]\)
\(\[ r^3 = \frac{2659.5}{\pi} \]\)
\(\[ r^3 \approx \frac{2659.5}{3.14159265359} \]\)
\(\[ r^3 \approx 846.484 \]\)
Now, to find the value of \(\( r \)\), we take the cube root of both sides:
\(\[ r \approx \sqrt[3]{846.484} \]\)
Calculating the cube root of 846.484 gives:
\(\[ r \approx 9.4519 \]\)
Lastly, to get the radius rounded to the nearest tenth of a centimeter, we round 9.4519 to one decimal place:
\(\[ r \approx 9.5 \text{ cm} \]\)
So the radius of the hemisphere, rounded to the nearest tenth of a centimeter, is 9.5 cm.
