At \(t\) minutes past 2 pm, the time needed by the minute hand of a clock to show 3 pm was found to be \(3\) minutes less than \(\dfrac{t^2}{4}\) minutes. Find \(t\).
\(t = 6\) minutes
Step 1: At 2 pm, the minute hand is at 12. At exactly 3 pm, the minute hand will come back to 12 again. So from 2 pm to 3 pm, there are 60 minutes in total.
Step 2: If it is already \(t\) minutes past 2 pm, then the time still left to reach 3 pm is \(60 - t\) minutes.
Step 3: According to the question, this remaining time is 3 minutes less than \(\dfrac{t^2}{4}\). So we can write the equation: \[ 60 - t = \dfrac{t^2}{4} - 3 \]
Step 4: Remove the fraction by multiplying everything by 4: \[ 4(60 - t) = t^2 - 12 \] Simplify: \[ 240 - 4t = t^2 - 12 \]
Step 5: Bring all terms to one side: \[ t^2 + 4t - 252 = 0 \]
Step 6: Factorise the quadratic: \[ (t - 6)(t + 42) = 0 \]
Step 7: So \(t = 6\) or \(t = -42\). But \(t\) represents minutes past 2 pm, so it must be between 0 and 60. Therefore, \(t = 6\).
Final Answer: The correct time is 6 minutes past 2 pm.