Question

\(f(t) = kt\) for all \(t\), where \(k\) is a constant, and \(f(3) = \frac{1}{2}\).
 

Column A	Column B
\(k\)	\(f(1)\)

Hint:

For any function of the form \(f(t) = kt\), the value of \(f(1)\) is always equal to \(k\), because \(f(1) = k \times 1 = k\). Recognizing this directly makes the comparison trivial without needing to calculate the specific value of k.

Answer 1

Step 1: Understanding the Concept:
This problem deals with a simple linear function, representing direct proportionality (\(f(t)\) is proportional to \(t\)). We are given a specific value of the function and asked to compare the constant of proportionality \(k\) with another value of the function.
Step 2: Key Formula or Approach:
First, use the given information \(f(3) = \frac{1}{2}\) to find the value of the constant \(k\). Then, use this value of \(k\) to evaluate the expression in Column B.
Step 3: Detailed Explanation:
The function is defined as \(f(t) = kt\).
We are \(f(3) = \frac{1}{2}\).
Using the function definition, we can write:
\[ f(3) = k \times 3 = 3k \] Since we know \(f(3) = \frac{1}{2}\), we can set up the equation:
\[ 3k = \frac{1}{2} \] To solve for \(k\), divide both sides by 3:
\[ k = \frac{1/2}{3} = \frac{1}{6} \] So, the value of Column A is \(\frac{1}{6}\).
Now for Column B, we need to find \(f(1)\).
Using the function definition again:
\[ f(1) = k \times 1 = k \] Since we have already found that \(k = \frac{1}{6}\), the value of Column B is also \(\frac{1}{6}\).
Step 4: Final Answer:
Comparing the two quantities:
Column A: \(k = \frac{1}{6}\)
Column B: \(f(1) = \frac{1}{6}\)
The two quantities are equal.