Step 1: Given Information
- Volume of the cone: \( V = 9856 \, \text{cm}^3 \) - Diameter of the base: \( d = 28 \, \text{cm} \) - Radius of the base: \( r = \frac{d}{2} = \frac{28}{2} = 14 \, \text{cm} \)
Step 2: Formula for Volume of the Cone
The formula for the volume \( V \) of a cone is: \[ V = \frac{1}{3} \pi r^2 h \] Substituting the known values \( V = 9856 \, \text{cm}^3 \) and \( r = 14 \, \text{cm} \), we can solve for the height \( h \). \[ 9856 = \frac{1}{3} \pi (14)^2 h \] \[ 9856 = \frac{1}{3} \pi \times 196 \times h \] \[ 9856 = \frac{1}{3} \times 3.1416 \times 196 \times h \] \[ 9856 = 205.76 h \] Solving for \( h \): \[ h = \frac{9856}{205.76} \approx 48 \, \text{cm} \] Therefore, the height of the cone is approximately \( 48 \, \text{cm} \).
Step 3: Formula for Slant Height of the Cone
The slant height \( l \) of the cone can be found using the Pythagorean theorem: \[ l = \sqrt{r^2 + h^2} \] Substituting \( r = 14 \, \text{cm} \) and \( h = 48 \, \text{cm} \): \[ l = \sqrt{14^2 + 48^2} = \sqrt{196 + 2304} = \sqrt{2500} = 50 \, \text{cm} \] Therefore, the slant height of the cone is \( 50 \, \text{cm} \).
Step 4: Formula for Curved Surface Area of the Cone
The formula for the curved surface area \( A \) of the cone is: \[ A = \pi r l \] Substituting \( r = 14 \, \text{cm} \) and \( l = 50 \, \text{cm} \): \[ A = \pi \times 14 \times 50 = 3.1416 \times 700 = 2199.12 \, \text{cm}^2 \] Therefore, the curved surface area of the cone is approximately \( 2199.12 \, \text{cm}^2 \).
The values for the cone are: