On a scale that measures the intensity of a certain phenomenon, a reading of n+1 corresponds to an intensity that...

1. Translate the problem requirements: The scale works like the Richter scale - each unit increase (\(\mathrm{n}\) to \(\mathrm{n+1}\)) means the intensity becomes 10 times greater. We need to find how many times greater the intensity at reading 8 is compared to reading 3.
2. Count the scale jumps: Determine how many unit increases occur when going from reading 3 to reading 8.
3. Apply the multiplication pattern: Since each unit increase multiplies intensity by 10, calculate the total multiplication factor for all the jumps.
4. Express in exponential form: Convert the result to match the answer choice format.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what this scale actually means in everyday terms. Think of it like the Richter scale for earthquakes - each time you go up one number on the scale, the actual intensity becomes dramatically stronger.

Specifically, this scale tells us that when the reading goes from any number \(\mathrm{n}\) to the next number \(\mathrm{n+1}\), the intensity becomes 10 times greater. So if we have a reading of 3 and it goes to 4, the intensity at 4 is 10 times the intensity at 3.

Our goal is clear: we want to find out how many times greater the intensity is at reading 8 compared to reading 3.

Process Skill: TRANSLATE - Converting the technical scale description into understandable language

2. Count the scale jumps

Now let's figure out how many steps we need to climb from reading 3 to reading 8.

Starting at 3 and going to 8:
\(3 \rightarrow 4\) (that's 1 jump)
\(4 \rightarrow 5\) (that's 2 jumps total)
\(5 \rightarrow 6\) (that's 3 jumps total)
\(6 \rightarrow 7\) (that's 4 jumps total)
\(7 \rightarrow 8\) (that's 5 jumps total)

So we have exactly 5 unit increases when going from reading 3 to reading 8.

We can also think of this more directly: \(8 - 3 = 5\) jumps.

3. Apply the multiplication pattern

Since each unit increase multiplies the intensity by 10, let's see what happens with our 5 jumps:

• After 1 jump (reading 4): intensity = 10 times the intensity at reading 3
• After 2 jumps (reading 5): intensity = \(10 \times 10 = 100\) times the intensity at reading 3
• After 3 jumps (reading 6): intensity = \(10 \times 10 \times 10 = 1,000\) times the intensity at reading 3
• After 4 jumps (reading 7): intensity = \(10 \times 10 \times 10 \times 10 = 10,000\) times the intensity at reading 3
• After 5 jumps (reading 8): intensity = \(10 \times 10 \times 10 \times 10 \times 10 = 100,000\) times the intensity at reading 3

So the intensity at reading 8 is 100,000 times as great as the intensity at reading 3.

4. Express in exponential form

We found that the intensity at reading 8 is 100,000 times the intensity at reading 3.

Let's express 100,000 in exponential form:
\(100,000 = 10 \times 10 \times 10 \times 10 \times 10 = 10^5\)

Looking at our answer choices, this matches choice (C) \(10^5\).

Final Answer

The intensity corresponding to a reading of 8 is \(10^5\) times as great as the intensity corresponding to a reading of 3.

Answer: (C) \(10^5\)

Verification: We moved 5 steps up the scale (from 3 to 8), and each step multiplies intensity by 10, giving us \(10^5 = 100,000\) times greater intensity.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the direction of the scale relationship

Students may incorrectly think that a reading of \(\mathrm{n+1}\) corresponds to an intensity that is 1/10 times (or 0.1 times) the intensity at reading \(\mathrm{n}\), reversing the multiplicative relationship. This happens because they might associate higher numbers with lower intensity, similar to some scales where smaller numbers indicate greater magnitude.

2. Confusing additive vs. multiplicative scaling

Students might think the scale works additively rather than multiplicatively. They may believe that going from reading 3 to reading 8 means the intensity increases by a factor of \((8-3) = 5\), rather than understanding that each unit increase multiplies the intensity by 10.

Errors while executing the approach

1. Incorrect counting of scale jumps

When counting from reading 3 to reading 8, students may count 6 jumps instead of 5. This happens because they count both endpoints (3, 4, 5, 6, 7, 8 = 6 numbers) rather than counting the number of steps between them (\(8-3 = 5\) steps).

2. Arithmetic errors in repeated multiplication

When calculating \(10 \times 10 \times 10 \times 10 \times 10\), students may lose track and either multiply the wrong number of times or make computational errors, arriving at values like \(10^4 = 10,000\) or \(10^6 = 1,000,000\) instead of the correct \(10^5 = 100,000\).

Errors while selecting the answer

1. Selecting the number of jumps instead of the exponential result

After correctly determining there are 5 jumps from reading 3 to reading 8, students might select answer choice (A) 5, thinking the answer is simply the number of scale steps rather than 10 raised to that power.