By measuring a 2-year-old's height, scientists can predict the child's height as an adult within 3 inches.

The formula used to determine the height is [tex]\( h = 2.2t + 4 \)[/tex], where [tex]\( t \)[/tex] is the 2-year-old's height in inches and [tex]\( h \)[/tex] is the height as an adult in inches.

If a 2-year-old's height is 29 inches, write and solve an absolute value inequality to determine the height range of the adult.



Answer :

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.