At what meter mark will Ario be when Miguel starts the race? Round to the nearest tenth.

[tex]\[ x = \left(\frac{m}{m+n}\right)\left(x_2 - x_1\right) + x_1 \][/tex]

Miguel and his brother Ario are both standing 3 meters from one side of a 25-meter pool when they decide to race. Miguel offers Ario a head start. Miguel says he will start when the ratio of Ario's completed meters to Ario's remaining meters is [tex]\(1: 4\)[/tex].

A. 4.4 meters
B. 7.4 meters
C. 17.6 meters
D. 20.6 meters



Answer :

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