Sure! Let's solve this problem step-by-step:
### (a) Finding the radius of the hemisphere
The total surface area of a hemisphere \(\( A \)\) is given as 618 cm². The formula for the total surface area of a hemisphere, including its base, is:
\(\[ A = 3 \pi r^2 \]\)
Where:
- \(\( A \)\) is the total surface area
- \(\( r \)\) is the radius of the hemisphere
- \(\( \pi \)\) is a mathematical constant approximately equal to 3.14159
Rearranging the formula to solve for \(\( r \)\):
\(\[ r^2 = \frac{A}{3 \pi} \]\)
Substituting 618 cm² for \(\( A \)\):
\(\[ r^2 = \frac{618}{3 \pi} \]\)
Now, taking the square root of both sides to solve for \(\( r \)\):
\(\[ r = \sqrt{\frac{618}{3 \pi}} \]\)
Using the correct numerical values, the radius \(\( r \)\) is found to be:
\(\[ r \approx 8.098 \, \text{cm} \]\)
### (b) Finding the volume of the hemisphere
The formula for the volume \(\( V \)\) of a hemisphere is:
\(\[ V = \frac{2}{3} \pi r^3 \]\)
Substituting the radius \(\( r \approx 8.098 \, \text{cm} \)\):
\(\[ V \approx \frac{2}{3} \pi (8.098)^3 \]\)
Calculating this, the volume \(\( V \)\) is approximately:
\(\[ V \approx 1112.076 \, \text{cm}^3 \]\)
### (c) Finding the external surface area of the hollow hemisphere
The external curved surface area of a hemisphere is given by the formula:
\(\[ A_{\text{curved}} = 2 \pi r^2 \]\)
Substituting the radius \(\( r \approx 8.098 \, \text{cm} \)\):
\(\[ A_{\text{curved}} \approx 2 \pi (8.098)^2 \]\)
Calculating this, the external curved surface area \(\( A_{\text{curved}} \)\) is approximately:
\(\[ A_{\text{curved}} \approx 412.0 \, \text{cm}^2 \]\)
To summarize:
(a) The radius of the hemisphere is approximately \(\( 8.098 \, \text{cm} \)\)
(b) The volume of the hemisphere is approximately \(\( 1112.076 \, \text{cm}^3 \)\)
(c) The external curved surface area of the hollow hemisphere is approximately \(\( 412.0 \, \text{cm}^2 \)\)
All answers are rounded to three significant figures.
