In another experiment, the height, \(y\) cm, of foam above a container \(t\) seconds after two chemicals are mixed is modelled by \(y = 12t - t^2\). Using t...
Question 1 Report
In another experiment, the height, \(y\) cm, of foam above a container \(t\) seconds after two chemicals are mixed is modelled by \(y = 12t - t^2\).
Using the fact that the foam reaches its maximum height when \(\dfrac{dy}{dt} = 0\), work out the maximum height of the foam. (3)Answer Details
This question uses differentiation to locate a maximum point on a curve, applying the fact that the gradient is zero at a turning point.
Differentiating \(y = 12t - t^2\):
\[\dfrac{dy}{dt} = 12 - 2t\]Setting the derivative to zero to find the turning point:
\[12 - 2t = 0 \Rightarrow t = 6\][2 marks]
Substituting \(t=6\) back into the original expression for \(y\):
\[y = 12(6) - 6^2 = 72 - 36 = 36 \text{ cm}\][1 mark]
Since the coefficient of \(t^2\) in the original expression is negative, the curve is an inverted parabola, so the turning point found is indeed a maximum, confirming that \(36\) cm is the greatest height the foam reaches.
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