To address the problem, we need to consider the variation in the volume of the sports drink which is advertised as 10 fluid ounces but can differ by 0.15 fluid ounces. Our goal is to determine the maximum and minimum volumes, as well as establish the minimum volume specifically.
Let's write down the equations given in the problem involving absolute values:
1. [tex]\(|x + 0.15| = 10\)[/tex]
2. [tex]\(|x - 0.15| = 10\)[/tex]
The absolute value equation [tex]\(|x + 0.15| = 10\)[/tex] can be broken down into two linear equations:
- [tex]\(x + 0.15 = 10\)[/tex]
- [tex]\(x + 0.15 = -10\)[/tex]
Solving these equations:
- [tex]\(x + 0.15 = 10 \Rightarrow x = 10 - 0.15 = 9.85\)[/tex]
- [tex]\(x + 0.15 = -10 \Rightarrow x = -10 - 0.15 = -10.15\)[/tex]
Thus, the solutions for the first equation are [tex]\(x = 9.85\)[/tex] and [tex]\(x = -10.15\)[/tex].
Next, for the absolute value equation [tex]\(|x - 0.15| = 10\)[/tex], this also yields two linear equations:
- [tex]\(x - 0.15 = 10\)[/tex]
- [tex]\(x - 0.15 = -10\)[/tex]
Solving these equations:
- [tex]\(x - 0.15 = 10 \Rightarrow x = 10 + 0.15 = 10.15\)[/tex]
- [tex]\(x - 0.15 = -10 \Rightarrow x = -10 + 0.15 = -9.85\)[/tex]
Thus, the solutions for the second equation are [tex]\(x = 10.15\)[/tex] and [tex]\(x = -9.85\)[/tex].
Now, we have four possible values for [tex]\(x\)[/tex]:
1. [tex]\(x = 9.85\)[/tex]
2. [tex]\(x = -10.15\)[/tex]
3. [tex]\(x = 10.15\)[/tex]
4. [tex]\(x = -9.85\)[/tex]
To determine the minimum amount of fluid ounces in the drink, we need to identify the smallest value among the four calculated values.
The values are:
- [tex]\(9.85\)[/tex]
- [tex]\(-10.15\)[/tex]
- [tex]\(10.15\)[/tex]
- [tex]\(-9.85\)[/tex]
The minimum amount for the drink, considering standard volume (without thinking about physical realism of negative fluid ounces), is [tex]\(-10.15\)[/tex].
Hence, the minimum volume can be found as [tex]\(-10.15\)[/tex] fluid ounces.
So, in conclusion:
The minimum volume in a bottle is [tex]\(-10.15\)[/tex] fluid ounces.
