Let's tackle this problem step-by-step.
### Given:
- Volume of the cone [tex]\( V = 107800 \, \text{cm}^3 \)[/tex]
- The ratio of the radius to the height of the cone is [tex]\( 5:12 \)[/tex]
### To Find:
1. Radius of the cone [tex]\( r \)[/tex]
2. Slant height of the cone [tex]\( l \)[/tex]
3. Height of the cone [tex]\( h \)[/tex]
4. Curved surface area of the cone
5. Total surface area of the cone
### Step-by-Step Solution:
#### Let the radius be [tex]\( r = 5x \)[/tex] and the height be [tex]\( h = 12x \)[/tex].
#### 1. Find the radius of the cone [tex]\( r \)[/tex]
We know that:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Substitute [tex]\( r = 5x \)[/tex] and [tex]\( h = 12x \)[/tex]:
[tex]\[ 107800 = \frac{1}{3} \pi (5x)^2 (12x) \][/tex]
Simplify:
[tex]\[ 107800 = \frac{1}{3} \pi 25x^2 12x \][/tex]
[tex]\[ 107800 = 100 \pi x^3 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x^3 = \frac{107800}{100 \pi} \][/tex]
[tex]\[ x = \left(\frac{107800}{100 \pi}\right)^{1/3} \][/tex]
Using the appropriate calculations, we find:
[tex]\[ x \approx 7.00093902326458 \][/tex]
Now, the radius [tex]\( r \)[/tex] is:
[tex]\[ r = 5x \approx 5 \times 7.00093902326458 \approx 35.00469519685729 \, \text{cm} \][/tex]
#### 2. Find the height of the cone [tex]\( h \)[/tex]
[tex]\[ h = 12x \approx 12 \times 7.00093902326458 \approx 84.0112684724575 \, \text{cm} \][/tex]
#### 3. Find the slant height of the cone [tex]\( l \)[/tex]
Using the Pythagorean Theorem for the right triangle formed by the radius, height, and slant height:
[tex]\[ l = \sqrt{r^2 + h^2} \][/tex]
[tex]\[ l = \sqrt{(35.00469519685729)^2 + (84.0112684724575)^2} \][/tex]
[tex]\[ l \approx \sqrt{1225.3281988594 + 7057.891576019842} \][/tex]
[tex]\[ l \approx 91.01220751182896 \, \text{cm} \][/tex]
#### 4. Find the curved surface area of the cone
The formula for the curved surface area [tex]\( A_{\text{curved}} \)[/tex] is:
[tex]\[ A_{\text{curved}} = \pi r l \][/tex]
Substitute the known values:
[tex]\[ A_{\text{curved}} \approx \pi \times 35.00469519685729 \times 91.01220751182896 \][/tex]
[tex]\[ A_{\text{curved}} \approx 10008.657353812756 \, \text{cm}^2 \][/tex]
#### 5. Find the total surface area of the cone
The total surface area [tex]\( A_{\text{total}} \)[/tex] is the sum of the base area and the curved surface area:
[tex]\[ A_{\text{total}} = \pi r^2 + \pi r l \][/tex]
First, calculate the base area:
[tex]\[ \text{Base area} = \pi r^2 \approx \pi \times (35.00469519685729)^2 \][/tex]
[tex]\[ \text{Base area} \approx \pi \times 1225.3281988594 \approx 3849.48359762029 \, \text{cm}^2 \][/tex]
Adding the curved surface area:
[tex]\[ A_{\text{total}} \approx 3849.48359762029 + 10008.657353812756 \][/tex]
[tex]\[ A_{\text{total}} \approx 13858.140951433046 \, \text{cm}^2 \][/tex]
### Summary of Results:
1. Radius [tex]\( r \approx 35.00469519685729 \, \text{cm} \)[/tex]
2. Slant height [tex]\( l \approx 91.01220751182896 \, \text{cm} \)[/tex]
3. Height [tex]\( h \approx 84.0112684724575 \, \text{cm} \)[/tex]
4. Curved surface area [tex]\( \approx 10008.657353812756 \, \text{cm}^2 \)[/tex]
5. Total surface area [tex]\( \approx 13858.140951433046 \, \text{cm}^2 \)[/tex]
