To determine the range of height that an adult might reach based on the height of a 2-year-old, we will use the given information and the formula provided. Let's go through the steps, one by one:
1. Determine the 2-year-old's height:
The height of the 2-year-old, [tex]\( t \)[/tex], is given as 29 inches.
2. Use the formula to predict adult height:
The formula provided to predict adult height [tex]\( h \)[/tex] based on the height of the 2-year-old [tex]\( t \)[/tex] is:
[tex]\[
h = 2.2t + 4
\][/tex]
3. Calculate the predicted height as an adult:
Substituting [tex]\( t = 29 \)[/tex] inches into the formula:
[tex]\[
h = 2.2(29) + 4
\][/tex]
Compute the value:
[tex]\[
h = 63.8 + 4
\][/tex]
[tex]\[
h = 67.8 \, \text{inches}
\][/tex]
4. Determine the height range within 3 inches:
The problem states that the predicted adult height is within 3 inches. Therefore, we set up an absolute value inequality:
[tex]\[
|h - 67.8| \le 3
\][/tex]
5. Convert the absolute value inequality into a compound inequality:
The inequality [tex]\( |h - 67.8| \le 3 \)[/tex] can be expressed as:
[tex]\[
-3 \le h - 67.8 \le 3
\][/tex]
6. Solve the compound inequality for [tex]\( h \)[/tex]:
Add 67.8 to all parts of the compound inequality:
[tex]\[
-3 + 67.8 \le h \le 3 + 67.8
\][/tex]
[tex]\[
64.8 \le h \le 70.8
\][/tex]
Therefore, based on a 2-year-old's height of 29 inches, the range of predicted adult height is from 64.8 inches to 70.8 inches. This can also be written in interval notation as:
[tex]\[
(64.8, 70.8)
\][/tex]
Thus, the height of the adult is expected to be between 64.8 inches and 70.8 inches.
