The side of a square is measured to be 12 inches with a possible error of ±0.02 inch in that measurement. If x is the exact length of the side of the square, express this difference as an absolute value inequality and solve. Then find the interval expressing the range of possible true areas of the square when using the measured length of 12 inches for a side.
To express the difference between the exact length of the side of the square (x) and the measured length (12 inches) as an absolute value inequality, one can use the given possible error of ±0.02 inch.
The absolute value of the difference between x and 12 should be less than or equal to the maximum possible error:
| x - 12 | ≤ 0.02
Now, let's solve this inequality:
To remove the absolute value, we consider two cases:
Case 1: x - 12 ≥ 0
In this case, the inequality becomes:
x - 12 ≤ 0.02
Solving for x:
x ≤ 12 + 0.02
x ≤ 12.02
Case 2: x - 12 < 0
In this case, the inequality becomes:
- (x - 12) ≤ 0.02 (negating the inequality)
-x + 12 ≤ 0.02
Solving for x:
x ≥ 12 - 0.02
x ≥ 11.98
Therefore, the solution to the absolute value inequality is:
11.98 ≤ x ≤ 12.02
This means the exact length of the side of the square (x) can range from 11.98 inches to 12.02 inches.
To find the range of possible true areas of the square when using the measured length of 12 inches for the side, we can utilize the equation A = s^2, where A is the area and s is the length of a side.
The minimum area occurs when s = 11.98 inches, and the maximum area occurs when s = 12.02 inches. Substituting these values into the equation:
Minimum area = (11.98)^2
Maximum area = (12.02)^2
Calculating these values:
Minimum area = 143.5204 square inches
Maximum area = 144.4804 square inches
Therefore, the range of possible true areas of the square, when using the measured length of 12 inches for a side, is from 143.5204 square inches to 144.4804 square inches.
11.98 < x < 12.02
It could be 12.02 *12.02 for example, or 11.98*11.98