# More MathPractice

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## Rates

Q Liquid is being poured into a parabolic bowl at a constant rate of ${\displaystyle 60\pi cm^{3}/s}$. The volume of the bowl is given by ${\displaystyle V=(1/2)\pi x^{4}}$, where the equation of the parabola is ${\displaystyle y=x^{2}}$, where y is the height of liquid in the bowl. Find the rate of increase of the height of the liquid in the bowl when the height is 10 centimetres.

A We can write Volume ${\displaystyle V=(1/2)\pi x^{4}=(1/2)\pi y^{2}}$

${\displaystyle {\frac {dV}{dt}}=\pi y{\frac {dy}{dt}}}$

${\displaystyle {\frac {dV}{dt}}}$ is ${\displaystyle 60\pi }$.

So, when y=10, we can evaluate ${\displaystyle {\frac {dy}{dt}}}$ which is the rate of change, which evaluates to be 6cm/s.