Certainly! Let's solve the given problem step by step:
### Given Data:
- Diameter of the base of the cone, [tex]\( d = 12 \)[/tex] cm
- Height of the cone, [tex]\( h = 14 \)[/tex] cm
- Value of [tex]\(\pi = \frac{27}{7} \)[/tex]
### Step 1: Calculate the Radius of the Cone
The radius ([tex]\( r \)[/tex]) of the base of the cone is half of its diameter.
[tex]\[ r = \frac{d}{2} = \frac{12 \, \text{cm}}{2} = 6 \, \text{cm} \][/tex]
### Step 2: Calculate the Volume of the Cone
The volume ([tex]\( V \)[/tex]) of a cone is given by the formula:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Substituting the given values:
[tex]\[ V = \frac{1}{3} \times \frac{27}{7} \times 6^2 \times 14 \][/tex]
[tex]\[ V = \frac{1}{3} \times \frac{27}{7} \times 36 \times 14 \][/tex]
[tex]\[ V = \frac{1}{3} \times \frac{27 \times 36 \times 14}{7} \][/tex]
After simplifying and calculating, we find:
[tex]\[ V = 648.0 \, \text{cm}^3 \][/tex]
### Step 3: Calculate the Slant Height of the Cone
The slant height ([tex]\( l \)[/tex]) of the cone is given by the Pythagorean theorem:
[tex]\[ l = \sqrt{r^2 + h^2} \][/tex]
Substituting the given values:
[tex]\[ l = \sqrt{6^2 + 14^2} \][/tex]
[tex]\[ l = \sqrt{36 + 196} \][/tex]
[tex]\[ l = \sqrt{232} \][/tex]
[tex]\[ l \approx 15.2 \, \text{cm} \][/tex]
### Step 4: Calculate the Curved Surface Area of the Cone
The curved surface area ([tex]\( A \)[/tex]) of the cone is given by the formula:
[tex]\[ A = \pi r l \][/tex]
Substituting the calculated and given values:
[tex]\[ A = \frac{27}{7} \times 6 \times 15.2 \][/tex]
[tex]\[ A = \frac{27 \times 6 \times 15.2}{7} \][/tex]
After simplifying and calculating, we find:
[tex]\[ A = 352.5 \, \text{cm}^2 \][/tex]
### Final Results:
1. Volume of the cone: [tex]\( 648.0 \, \text{cm}^3 \)[/tex]
2. Curved Surface Area of the cone: [tex]\( 352.5 \, \text{cm}^2 \)[/tex]
These calculations provide the required volume and curved surface area of the cone to 1 decimal place.
