If `k(x)=4x^(3)a, and k(3)=27`, what is the k(2)?

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the function and given information
We are given the function:
\[ k(x) = 4x^3 a \]
and the information that:
\[ k(3) = 27 \]

### Step 2: Substitute \( x = 3 \) into the function
To find the value of \( a \), we substitute \( x = 3 \) into the function:
\[ k(3) = 4(3^3)a \]
Calculating \( 3^3 \):
\[ 3^3 = 27 \]
Thus, we have:
\[ k(3) = 4 \cdot 27 \cdot a = 108a \]

### Step 3: Set the equation equal to 27
Since we know that \( k(3) = 27 \), we can set up the equation:
\[ 108a = 27 \]

### Step 4: Solve for \( a \)
To find \( a \), we divide both sides of the equation by 108:
\[ a = \frac{27}{108} \]
Simplifying:
\[ a = \frac{1}{4} \]

### Step 5: Substitute \( a \) back into the function
Now that we have found \( a \), we substitute it back into the original function:
\[ k(x) = 4x^3 \left(\frac{1}{4}\right) \]
The \( 4 \) cancels out:
\[ k(x) = x^3 \]

### Step 6: Find \( k(2) \)
Now we need to find \( k(2) \):
\[ k(2) = 2^3 \]
Calculating \( 2^3 \):
\[ 2^3 = 8 \]

### Conclusion
Thus, the value of \( k(2) \) is:
\[ \boxed{8} \]

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