Alessandra is raising turtles and must keep the air temperature in their enclosure at [tex]\( 34^{\circ} C \)[/tex], with a relative error less than [tex]\( 4 \% \)[/tex]. Write an absolute value inequality for the actual temperature. Find the interval in which the actual temperature must lie.

Write the absolute value inequality for the actual temperature:
[tex]\[
\frac{|t - 34|}{34} \ \textless \ 0.04
\][/tex]
(Type an inequality using [tex]\( t \)[/tex] as the variable. Use integers or decimals for any numbers in the inequality. Do not include the degree symbol in your answers.)

Therefore, the temperature must lie in the interval [tex]\( \square \)[/tex]
(Type your answer in interval notation. Type an integer or a decimal. Do not include the degree symbol in your answer.)



Answer :

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]