If x is a number such that 0.020 which of the following is closest to the value of (1/(sqrt(10)) -...

1. Translate the problem requirements: We have \(\mathrm{x}\) close to \(\sqrt{10}\) (within 0.020 to 0.021 above \(\sqrt{10}\)) and need to find the approximate value of \(\frac{1}{\sqrt{10}} - \frac{1}{\mathrm{x}}\). This is asking how the difference between these two reciprocals changes when \(\mathrm{x}\) is slightly larger than \(\sqrt{10}\).
2. Establish the relationship between x and √10: Use the given inequality to express \(\mathrm{x}\) in terms of \(\sqrt{10}\) plus a small amount, which will help us work with concrete values.
3. Apply algebraic manipulation to simplify the expression: Transform \(\frac{1}{\sqrt{10}} - \frac{1}{\mathrm{x}}\) into a form that's easier to calculate by finding a common denominator and factoring.
4. Use approximation based on answer choices: Since answer choices are orders of magnitude apart, we can use \(\sqrt{10} \approx 3\) and approximate \(\mathrm{x}\) to get a quick, accurate estimate without complex calculations.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is actually asking us. We're told that \(\mathrm{x}\) is very close to \(\sqrt{10}\), specifically that \(\mathrm{x}\) is between 0.020 and 0.021 units larger than \(\sqrt{10}\). Think of it this way: if \(\sqrt{10}\) is like a target, then \(\mathrm{x}\) is hitting just slightly to the right of that target.

We need to find the value of \(\frac{1}{\sqrt{10}} - \frac{1}{\mathrm{x}}\). This expression is asking: "How much bigger is the reciprocal of \(\sqrt{10}\) compared to the reciprocal of \(\mathrm{x}\)?" Since \(\mathrm{x}\) is slightly larger than \(\sqrt{10}\), we expect \(\frac{1}{\mathrm{x}}\) to be slightly smaller than \(\frac{1}{\sqrt{10}}\), so our answer should be a small positive number.

Process Skill: TRANSLATE - Converting the inequality constraint into understanding about the relative positions of x and √10

2. Establish the relationship between x and √10

From the given inequality \(0.020 < \mathrm{x} - \sqrt{10} < 0.021\), we can rewrite this as:
\(\sqrt{10} + 0.020 < \mathrm{x} < \sqrt{10} + 0.021\)

To make our work easier, let's use the fact that \(\sqrt{10} \approx 3.162\). So \(\mathrm{x}\) is approximately:
\(3.162 + 0.020 < \mathrm{x} < 3.162 + 0.021\)
\(3.182 < \mathrm{x} < 3.183\)

For our calculations, we can use \(\mathrm{x} \approx 3.182\) as a representative value. This gives us concrete numbers to work with instead of keeping everything in abstract terms.

3. Apply algebraic manipulation to simplify the expression

Now let's work with \(\frac{1}{\sqrt{10}} - \frac{1}{\mathrm{x}}\). To subtract these fractions, we need a common denominator:

\(\frac{1}{\sqrt{10}} - \frac{1}{\mathrm{x}} = \frac{\mathrm{x} - \sqrt{10}}{\mathrm{x} \cdot \sqrt{10}}\)

This is a powerful simplification! Notice that the numerator \(\mathrm{x} - \sqrt{10}\) is exactly what we know from our constraint: it's between 0.020 and 0.021.

So our expression becomes: \(\frac{\mathrm{x} - \sqrt{10}}{\mathrm{x} \cdot \sqrt{10}} \approx \frac{0.0205}{\mathrm{x} \cdot \sqrt{10}}\)

Using \(\mathrm{x} \approx 3.182\) and \(\sqrt{10} \approx 3.162\):
Denominator = \(\mathrm{x} \cdot \sqrt{10} \approx 3.182 \times 3.162 \approx 10.06\)

Process Skill: MANIPULATE - Transforming the original expression into a form that directly uses our constraint

4. Use approximation based on answer choices

Now we can calculate: \(\frac{\mathrm{x} - \sqrt{10}}{\mathrm{x} \cdot \sqrt{10}} \approx \frac{0.0205}{10.06} \approx 0.002\)

Let's verify this makes sense: since \(\mathrm{x}\) is very close to \(\sqrt{10}\), the product \(\mathrm{x} \cdot \sqrt{10}\) is very close to \((\sqrt{10})^2 = 10\). So we're essentially calculating \(\frac{0.020}{10} = 0.002\).

Looking at our answer choices: 0.00002, 0.0002, 0.002, 0.02, 0.2

Our calculated value of 0.002 matches choice C exactly.

Final Answer

The value of \(\frac{1}{\sqrt{10}} - \frac{1}{\mathrm{x}}\) is closest to 0.002.

Answer: C

Verification: Our algebraic manipulation showed that the expression equals \(\frac{\mathrm{x} - \sqrt{10}}{\mathrm{x} \cdot \sqrt{10}}\), where \(\mathrm{x} - \sqrt{10} \approx 0.021\) and \(\mathrm{x} \cdot \sqrt{10} \approx 10\), giving us approximately \(\frac{0.021}{10} = 0.002\).

Common Faltering Points

Errors while devising the approach

Students often misread the given inequality \(0.020 < \mathrm{x} - \sqrt{10} < 0.021\) as \(0.020 < \mathrm{x} < 0.021\), completely missing the "\(- \sqrt{10}\)" part. This leads them to think \(\mathrm{x}\) is a very small number between 0.020 and 0.021, rather than understanding that \(\mathrm{x}\) is actually close to \(\sqrt{10} \approx 3.162\). This fundamental misunderstanding derails the entire solution from the start.

Many students try to directly substitute approximate values into \(\frac{1}{\sqrt{10}} - \frac{1}{\mathrm{x}}\) without first simplifying the expression algebraically. They might calculate \(\frac{1}{\sqrt{10}} \approx 0.316\) and \(\frac{1}{\mathrm{x}} \approx 0.314\), then subtract to get 0.002. While this might work, it's prone to rounding errors and doesn't reveal the elegant relationship between the constraint and the target expression.

Students may not realize that the numerator in the simplified form \(\frac{\mathrm{x} - \sqrt{10}}{\mathrm{x} \cdot \sqrt{10}}\) is exactly what's given in the constraint. They treat the constraint as separate information rather than recognizing it as the key to solving the problem efficiently.

Errors while executing the approach

When converting \(\frac{1}{\sqrt{10}} - \frac{1}{\mathrm{x}}\) to a single fraction, students commonly make errors like writing \(\frac{\sqrt{10} - \mathrm{x}}{\mathrm{x} \cdot \sqrt{10}}\) instead of \(\frac{\mathrm{x} - \sqrt{10}}{\mathrm{x} \cdot \sqrt{10}}\). This sign error completely changes the problem since it makes the result negative when it should be positive.

Students might use rough approximations like \(\sqrt{10} \approx 3\) instead of the more accurate \(\sqrt{10} \approx 3.162\). When dealing with small differences, this level of approximation error can lead to selecting the wrong answer choice, especially when distinguishing between 0.002 and 0.02.

When calculating \(\frac{0.0205}{\mathrm{x} \cdot \sqrt{10}} \approx \frac{0.0205}{10.06}\), students may make basic division errors or incorrectly handle decimal places, potentially getting results like 0.02 or 0.0002 instead of the correct 0.002.

Errors while selecting the answer

Students might reason that since \(\mathrm{x} - \sqrt{10}\) is around 0.02 and both \(\mathrm{x}\) and \(\sqrt{10}\) are around 3, the answer should be roughly \(\frac{0.02}{3} \approx 0.007\), leading them to incorrectly select 0.02 as the closest answer choice instead of recognizing that 0.002 is actually closer to their estimate.

With answer choices ranging from 0.00002 to 0.2, students can easily miscount decimal places, especially under time pressure. They might calculate 0.002 correctly but then select 0.0002 (choice B) or 0.02 (choice D) due to decimal place confusion.