To determine at what meter mark Ario will be when Miguel starts the race, we need to use the given formula:
[tex]\[
x = \left(\frac{m}{m+n}\right) (x_2 - x_1) + x_1
\][/tex]
where:
- [tex]\(x_1 = 3\)[/tex] meters (the starting point for both Miguel and Ario)
- [tex]\(x_2 = 25\)[/tex] meters (the end of the pool)
- [tex]\(m\)[/tex] is the part of the race Ario has completed
- [tex]\(n\)[/tex] is the part of the race Ario still has to complete
Given that the ratio of Ario's completed meters to his remaining meters is [tex]\(1: 4\)[/tex], we have [tex]\(m = 1\)[/tex] and [tex]\(n = 4\)[/tex].
Now, we need to substitute these values into the formula.
First, calculate the fraction:
[tex]\[
\frac{m}{m+n} = \frac{1}{1+4} = \frac{1}{5}
\][/tex]
Next, compute the meter mark using the formula:
[tex]\[
x = \left(\frac{1}{5}\right)(x_2 - x_1) + x_1
\][/tex]
Substituting [tex]\(x_2 = 25\)[/tex] and [tex]\(x_1 = 3\)[/tex]:
[tex]\[
x = \left(\frac{1}{5}\right)(25 - 3) + 3
\][/tex]
Simplify inside the parentheses:
[tex]\[
x = \left(\frac{1}{5}\right)(22) + 3
\][/tex]
Then multiply:
[tex]\[
x = 4.4 + 3
\][/tex]
Finally, add the values:
[tex]\[
x = 7.4
\][/tex]
Thus, Ario will be at the 7.4 meter mark when Miguel starts the race. Therefore, the answer is:
[tex]\[
7.4 \text{ meters}
\][/tex]
So, the correct option is:
7.4 meters
