Question

calculate the radius of the given hemi-sphere which has a volume of 1111 cm³. use π = 3.142.

Explanation:

Step1: Recall the volume formula for a hemisphere

The volume \( V \) of a hemisphere is given by the formula \( V=\frac{2}{3}\pi r^{3} \), where \( r \) is the radius of the hemisphere. We know that \( V = 1111\space cm^{3}\) and \( \pi=3.142 \). We need to solve for \( r \).

First, substitute the known values into the formula:
\( 1111=\frac{2}{3}\times3.142\times r^{3} \)

Step2: Simplify the right - hand side of the equation

First, calculate \( \frac{2}{3}\times3.142=\frac{6.284}{3}\approx2.0947 \)
So the equation becomes \( 1111 = 2.0947\times r^{3} \)

Step3: Solve for \( r^{3} \)

To find \( r^{3} \), we divide both sides of the equation by \( 2.0947 \):
\( r^{3}=\frac{1111}{2.0947}\approx530.4 \)

Step4: Solve for \( r \)

To find \( r \), we take the cube root of \( r^{3} \). So \( r=\sqrt[3]{530.4} \)
Calculating the cube root, we know that \( 8^{3}=512 \) and \( 8.1^{3}=8.1\times8.1\times8.1 = 65.61\times8.1=531.441 \)
Since \( 530.4 \) is very close to \( 531.441 \), we can approximate \( r\approx8.1\space cm \) (we can also use a calculator for more precise calculation: \( \sqrt[3]{530.4}\approx8.1 \))

Answer:

The radius of the hemisphere is approximately \( 8.1\space cm \)