In the formula w = {p}{sqrt[t]{v}}, integers p and t are positive constants. If w = 2 when v =...

1. Translate the problem requirements: We have a formula \(\mathrm{w = p/\sqrt[t]{v}}\) where p and t are positive integer constants. We're given two conditions: when \(\mathrm{v=1, w=2}\), and when \(\mathrm{v=64, w=1/2}\). We need to find the value of t.
2. Use the first condition to find p: Substitute the simpler condition \(\mathrm{(v=1, w=2)}\) into the formula to determine the constant p, since \(\mathrm{\sqrt[t]{1} = 1}\) regardless of t value.
3. Apply the second condition with known p: Use the second condition \(\mathrm{(v=64, w=1/2)}\) along with the value of p found in step 2 to create an equation involving only t.
4. Solve for t by recognizing power relationships: Manipulate the equation to identify what power of 64 gives us the required result, remembering that 64 is a perfect power of smaller integers.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have in plain English. We're given a formula that shows how three quantities relate to each other: w, p, v, and t. Think of it like a recipe where p and t are fixed ingredients (positive integer constants), and we can change v to get different values of w.

The formula is \(\mathrm{w = p/\sqrt[t]{v}}\), where the symbol ∛ means "take the t-th root of v".

We're told two specific situations:

- When \(\mathrm{v = 1}\), then \(\mathrm{w = 2}\)
- When \(\mathrm{v = 64}\), then \(\mathrm{w = 1/2}\)

Our goal is to find what value of t makes both of these situations true.

Process Skill: TRANSLATE

2. Use the first condition to find p

Let's use the simpler situation first to find p. When \(\mathrm{v = 1}\) and \(\mathrm{w = 2}\), we can substitute these into our formula.

Here's the key insight: no matter what value t has, when we take the t-th root of 1, we always get 1. This is because 1 raised to any power equals 1.

So when \(\mathrm{v = 1}\):
\(\mathrm{\sqrt[t]{1} = 1}\) (regardless of what t is)

Substituting into our formula:
\(\mathrm{2 = p/1}\)
Therefore: \(\mathrm{p = 2}\)

Now we know that \(\mathrm{p = 2}\), so our formula becomes: \(\mathrm{w = 2/\sqrt[t]{v}}\)

3. Apply the second condition with known p

Now let's use our second condition. We know that when \(\mathrm{v = 64}\), \(\mathrm{w = 1/2}\), and we just found that \(\mathrm{p = 2}\).

Substituting into our formula:
\(\mathrm{1/2 = 2/\sqrt[t]{64}}\)

To solve this, let's think about what this equation is telling us. We need:
\(\mathrm{\sqrt[t]{64} = 2 ÷ (1/2) = 2 × 2 = 4}\)

So we need: \(\mathrm{\sqrt[t]{64} = 4}\)

In other words, we need to find what value of t makes the t-th root of 64 equal to 4.

Process Skill: MANIPULATE

4. Solve for t by recognizing power relationships

Now we need to figure out: what power of 4 gives us 64?

Let's think about this step by step:

- \(\mathrm{4^1 = 4}\)
- \(\mathrm{4^2 = 16}\)
- \(\mathrm{4^3 = 64}\)

Perfect! We found that \(\mathrm{4^3 = 64}\).

This means that the 3rd root of 64 equals 4, or in mathematical notation: \(\mathrm{\sqrt[3]{64} = 4}\).

Therefore, \(\mathrm{t = 3}\).

Let's verify: If \(\mathrm{t = 3}\), then our formula is \(\mathrm{w = 2/\sqrt[3]{v}}\)

- When \(\mathrm{v = 1}\): \(\mathrm{w = 2/\sqrt[3]{1} = 2/1 = 2}\) ✓
- When \(\mathrm{v = 64}\): \(\mathrm{w = 2/\sqrt[3]{64} = 2/4 = 1/2}\) ✓

Both conditions are satisfied!

Final Answer

The value of \(\mathrm{t = 3}\), which corresponds to answer choice (C).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the root notation: Students may confuse the notation \(\mathrm{\sqrt[t]{v}}\) (t-th root of v) with a cube root specifically, assuming \(\mathrm{t = 3}\) from the start instead of recognizing that t is the unknown variable we need to find.

2. Wrong order of using conditions: Students might attempt to use the more complex condition \(\mathrm{(v = 64, w = 1/2)}\) first instead of starting with the simpler condition \(\mathrm{(v = 1, w = 2)}\) to find p. This makes the algebra unnecessarily complicated.

3. Forgetting that p and t are constants: Students may treat p as a variable that changes between the two conditions, not realizing that p must remain the same constant value in both scenarios.

Errors while executing the approach

1. Arithmetic errors with powers and roots: When calculating \(\mathrm{4^3 = 64}\), students might make computational mistakes or confuse the relationship between powers and roots, such as thinking \(\mathrm{4^2 = 64}\) instead of \(\mathrm{4^3 = 64}\).

2. Algebraic manipulation errors: When solving \(\mathrm{1/2 = 2/\sqrt[t]{64}}\), students may incorrectly cross-multiply or make errors in isolating \(\mathrm{\sqrt[t]{64}}\), potentially getting \(\mathrm{\sqrt[t]{64} = 1}\) instead of \(\mathrm{\sqrt[t]{64} = 4}\).

3. Incorrect handling of the root property: Students might not recognize that \(\mathrm{\sqrt[t]{1} = 1}\) for any value of t, leading to unnecessarily complex calculations when using the first condition.

Errors while selecting the answer

1. Confusing the value of t with other calculated values: Students might select \(\mathrm{p = 2}\) or \(\mathrm{\sqrt[t]{64} = 4}\) as their final answer instead of \(\mathrm{t = 3}\), mixing up which variable the question is asking for.

2. Not verifying the answer: Students may arrive at \(\mathrm{t = 3}\) but fail to check both conditions, potentially missing calculation errors that would make them reconsider their answer choice.