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