T varies inversely as the cube of R. When R = 3, T = $\frac{2}{81}$, find T when R = 2

1. Analyze the question

We are given a relationship of inverse variation: $T$ varies inversely as the cube of $R$. We are provided with an initial set of values ($R=3, T=\frac{2}{81}$) and asked to find the value of $T$ when $R=2$.

2. Explain the concept

In inverse variation, the relationship is expressed as:

To turn this into an equation, we introduce a constant of proportionality, $k$:

Our goal is to first find the value of $k$ using the given values, and then use that $k$ to solve for the unknown $T$.

3. Show the solution

Step 1: Find the constant $k$ Using the values $T = \frac{2}{81}$ and $R = 3$:

Now, solve for $k$ by multiplying both sides by 27:

Since $81 = 27 \times 3$, we can simplify:

Step 2: Find $T$ when $R = 2$ Now that we know $k = \frac{2}{3}$, our formula is:

Substitute $R = 2$:

4. Identify the answer

Comparing our result to the given options: A. $\frac{1}{18}$ B. $\frac{1}{12}$ C. $\frac{1}{24}$ D. $\frac{1}{6}$

The correct option is B.

5. Tip

When dealing with inverse variation, always remember the "Product Rule": if $T \propto \frac{1}{R^3}$, then $T_1 \times R_1^3 = T_2 \times R_2^3$. This allows you to solve the problem without explicitly finding $k$ if you prefer!