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