A lab technician records the temperature, \(T\) degrees Celsius, of a sample sitting on a hotplate, \(t\) minutes after heating begins, as \(T = t^{2} - 8t ...
Question 1 Report
A lab technician records the temperature, \(T\) degrees Celsius, of a sample sitting on a hotplate, \(t\) minutes after heating begins, as \(T = t^{2} - 8t + 20\).
- Express \(T\) in the form \((t-p)^{2}+q\), where \(p\) and \(q\) are constants. (3)
- Hence state the minimum temperature reached and the value of \(t\) at which it occurs. (2)
Answer Details
Completing the square rewrites a quadratic as \((t-p)^{2}+q\); because a squared term can never be negative, this form immediately reveals the minimum value of the expression and where it occurs.
(a) Halving the coefficient of \(t\) gives \(p=4\): \(T = t^{2}-8t+20 = (t-4)^{2}-4^{2}+20 = (t-4)^{2}-16+20 = (t-4)^{2}+4\). [3 marks]
(b) Since \((t-4)^{2}\geqslant0\) for every value of \(t\), the smallest possible value of \(T\) occurs when \((t-4)^{2}=0\), i.e. when \(t=4\) minutes, giving a minimum temperature of \(4^{\circ}\)C. [2 marks]
Exam tip: the constant added after completing the square (here 4) is always the minimum value for a positive squared term, and the number subtracted inside the bracket (here 4) is always where that minimum occurs.
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