Since we need to find the radius, height, and volume of the cone, let's proceed step-by-step.
(i) Finding the radius of the cone:
The formula to calculate the curved surface area [tex]\( A \)[/tex] of a cone in terms of its radius [tex]\( r \)[/tex] and slant height [tex]\( l \)[/tex] is:
[tex]\[ A = \pi r l \][/tex]
Given:
[tex]\[ A = 550 \, \text{cm}^2 \][/tex]
[tex]\[ l = 25 \, \text{cm} \][/tex]
We can rearrange the formula to solve for the radius [tex]\( r \)[/tex]:
[tex]\[ r = \frac{A}{\pi l} \][/tex]
Substituting in the given values:
[tex]\[ r = \frac{550}{\pi \times 25} \][/tex]
Calculating the radius:
[tex]\[ r \approx 7.0028 \, \text{cm} \][/tex]
(ii) Finding the height of the cone:
We use the Pythagorean theorem for the right triangle formed by the radius [tex]\( r \)[/tex], the height [tex]\( h \)[/tex], and the slant height [tex]\( l \)[/tex]. The relationship is given by:
[tex]\[ l^2 = r^2 + h^2 \][/tex]
Rearrange to solve for the height [tex]\( h \)[/tex]:
[tex]\[ h = \sqrt{l^2 - r^2} \][/tex]
Substituting in the values:
[tex]\[ h = \sqrt{25^2 - 7.0028^2} \][/tex]
Calculating the height:
[tex]\[ h \approx 23.9992 \, \text{cm} \][/tex]
(iii) Finding the volume of the cone:
The volume [tex]\( V \)[/tex] of a cone is given by:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Using the values for the radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex] from above:
[tex]\[ V = \frac{1}{3} \pi \times (7.0028)^2 \times 23.9992 \][/tex]
Calculating the volume:
[tex]\[ V \approx 1232.454 \, \text{cm}^3 \][/tex]
To summarize:
- The radius of the cone is approximately [tex]\( 7.0028 \, \text{cm} \)[/tex].
- The height of the cone is approximately [tex]\( 23.9992 \, \text{cm} \)[/tex].
- The volume of the cone is approximately [tex]\( 1232.454 \, \text{cm}^3 \)[/tex].
