Problem 60 - Calculus Single Variable

Question

Determine the interval that \(y\) must lie in to agree with \(x\) to \(q\) decimal places. \(x=24, q=2\)

Step-by-Step Solution

Answer

The interval is \([23.995, 24.005)\).

Step 1: Define the Agreement Condition

To agree to two decimal places, the value of \(y\) must round to the same number as \(x\) when both are rounded to two decimal places. Since \(x = 24\), its value remains \(24.00\) when extended to two decimal places.

Step 2: Understand Rounding Rules

A number \(x\) rounded to two decimal places will lie within \(0.005\) units above and below. Therefore, 2 decimal place rounding means considering the number is precisely accurate between \(x - 0.005\) and \(x + 0.005\).

Step 3: Calculate the Exact Interval

Since we need the interval for two decimal places, the number rounded is \(24.00\). The lower boundary is \(24.00 - 0.005 = 23.995\) and the upper boundary is \(24.00 + 0.005 = 24.005\). Thus, \(y\) must lie within the interval \([23.995, 24.005)\).

Key Concepts

Understanding the Agreement Condition

When we talk about the **agreement condition** in mathematics, particularly with decimal places, we refer to the requirement that two numbers agree or match at a certain level of precision, such as to the same number of decimal places. In the context of the problem, this means that the number\( y \) should match the number\( x \) when both are rounded to a specified number of decimal places, in this case, two decimal places.
To illustrate, if \( x = 24 \), it rounds to \( 24.00 \) when expressed to two decimal places. Consequently, \( y \) must also round to \( 24.00 \) to meet the agreement condition. This concept ensures accuracy and precision in mathematical calculations and comparisons.

Mastering Rounding Rules

Rounding rules are essential when working with decimals. They help determine how a number is approximated when precision is reduced. To round a number to two decimal places, consider the digit in the third decimal place:

• If the third decimal is 5 or more, round the second decimal place up.
• If the third decimal is less than 5, the second decimal place remains unchanged.

In the provided exercise, since \( x = 24 \), rounding it to two decimal places simply converts it to \( 24.00 \). The rounding rules are applied here to determine if changes are required beyond the second decimal digit, maintaining accuracy within a small range.
The critical takeaway from rounding rules is to understand how they create a boundary or range around our number, preparing us for the next step: interval calculation.

Precision and Interval Calculation

Interval calculation defines the range within which the value \( y \) should lie to satisfy the agreement condition with \( x \) to two decimal places. After rounding, both numbers must fall within a certain close proximity to be considered the same for precision at the defined number of decimal places.
For our example, we need to calculate a range around the rounded number \( 24.00 \). This range is often determined by the value of \( 0.005 \), which accounts for half of the unit in the third decimal place:

• Lower boundary: \( 24.00 - 0.005 = 23.995 \)
• Upper boundary: \( 24.00 + 0.005 = 24.005 \)

Thus, to agree with \( x \) at two decimal places, \( y \) must fall within the interval \([23.995, 24.005)\). Understanding this precision range is crucial for ensuring that numbers align correctly in mathematical computations.