### Step-by-Step Solution
#### Part 1: Sum of Interior Angles of a Polygon with 18 Sides
To find the sum of interior angles of a polygon with [tex]\( n \)[/tex] sides, you can use the formula:
[tex]\[ \text{Sum of interior angles} = 180^\circ (n - 2) \][/tex]
Given:
- Number of sides, [tex]\( n = 18 \)[/tex]
Let's plug in the value:
[tex]\[ \text{Sum of interior angles} = 180^\circ (18 - 2) = 180^\circ \times 16 = 2880^\circ \][/tex]
So, the sum of interior angles of a polygon with 18 sides is [tex]\( 2880^\circ \)[/tex].
#### Part 2: Finding the Son's Age and Father's Age
Given conditions:
1. The father's age is 24 times the son's age.
2. In 4 years, the ratio of their ages will be 4:1.
Let's denote the son's current age as [tex]\( x \)[/tex].
Then, the father's current age will be [tex]\( 24x \)[/tex].
In 4 years:
- The son's age will be [tex]\( x + 4 \)[/tex]
- The father's age will be [tex]\( 24x + 4 \)[/tex]
According to the problem, the ratio of the father's age to the son's age in 4 years will be 4:1.
Setting up the equation for the ratio:
[tex]\[ \frac{24x + 4}{x + 4} = 4 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
1. Cross-multiply to get rid of the fraction:
[tex]\[ 24x + 4 = 4(x + 4) \][/tex]
2. Distribute on the right side:
[tex]\[ 24x + 4 = 4x + 16 \][/tex]
3. Move all terms involving [tex]\( x \)[/tex] to one side and constants to the other side:
[tex]\[ 24x - 4x = 16 - 4 \][/tex]
[tex]\[ 20x = 12 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{12}{20} = \frac{3}{5} \][/tex]
So, the son's current age is [tex]\( \frac{3}{5} \)[/tex] years.
To find the father's current age:
[tex]\[ \text{Father's age} = 24 \times \frac{3}{5} = \frac{72}{5} \][/tex]
Thus, the father's current age is [tex]\( \frac{72}{5} \)[/tex] years.
### Summary
- The sum of the interior angles of an 18-sided polygon is [tex]\( 2880^\circ \)[/tex].
- The son's current age is [tex]\( \frac{3}{5} \)[/tex] years.
- The father's current age is [tex]\( \frac{72}{5} \)[/tex] years.
