Alessandra is raising turtles and must ensure that the air temperature in their enclosure is maintained at [tex]\(34^\circ C\)[/tex], with a relative error of less than [tex]\(4\%\)[/tex].
First, we need to translate the given information into an absolute value inequality.
The problem gives us the desired temperature and the relative error:
- Desired temperature: [tex]\(34^\circ C\)[/tex]
- Relative error: [tex]\(4\%\)[/tex]
Using this information, we can create the absolute value inequality:
[tex]\[
\frac{|t - 34|}{34} < 0.04
\][/tex]
Next, to simplify this inequality, we start by eliminating the fraction. We do this by multiplying both sides of the inequality by 34:
[tex]\[
|t - 34| < 0.04 \times 34
\][/tex]
Solving the multiplication on the right side, we get:
[tex]\[
|t - 34| < 1.36
\][/tex]
This inequality tells us that the temperature [tex]\(t\)[/tex] must be within [tex]\(1.36^\circ C\)[/tex] of [tex]\(34^\circ C\)[/tex].
To express this as a standard absolute value inequality, we can rewrite it as:
[tex]\[
-1.36 < t - 34 < 1.36
\][/tex]
Next, we will isolate [tex]\(t\)[/tex] by adding 34 to all parts of the inequality:
[tex]\[
-1.36 + 34 < t < 1.36 + 34
\][/tex]
Simplifying the addition, we find:
[tex]\[
32.64 < t < 35.36
\][/tex]
Therefore, the interval in which the actual temperature must lie is:
[tex]\[
(32.64, 35.36)
\][/tex]
So the required answers are:
- The absolute value inequality: [tex]\(\frac{|t - 34|}{34} < 0.04\)[/tex]
- The interval notation: [tex]\((32.64, 35.36)\)[/tex]
