Rates of Change involving Integration

Rates of Change for finding height and time can be solved by integration. Once integrated, find the integral constant using the given conditions.

Worked Examples for Rates of Change involving Integration

The diagram represents a vertical cylindrical water cooler of constant cross-sectional area \(A\). Water drains through a hole at the bottom of the cooler. From physical principles, it is known that the volume \(V\) of water decreases at a rate given by
$$ \displaystyle \frac{dV}{dt} = -k \sqrt{y}, $$
where \(k\) is a positive constant and \(y\) is the depth of water.
Initially, the cooler is full and takes \(T\) seconds to drain. Thus \(y = y_0 \) when \(t=0\), and \(y=0\) when \(t=T\).

(a)    Show that \(\displaystyle \frac{dy}{dt} = -\frac{k}{A} \sqrt{y} \).

\( \begin{aligned} \displaystyle \require{color}
V &= \text{cross-sectional area } \times \text{ height} \\
&= Ah \\
\frac{dy}{dt} &= \frac{dy}{dV} \times \frac{dV}{dt} \\
&= \frac{1}{A} \times -k \sqrt{y} \\
\therefore \frac{dy}{dt} &= – \frac{k}{A} \sqrt{y}
\end{aligned} \)

(b)    Show that \( \displaystyle y = y_0 \bigg(1-\frac{t}{T}\bigg)^2 \text{ for } 0 \le t \le T \).

\( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}
\frac{dt}{dy} &= -\frac{A}{k} \frac{1}{\sqrt{y}} &\color{red} \text{by } (a) \\
t &= -\frac{A}{k} \int \frac{1}{\sqrt{y}} dy \\
&= -\frac{2A}{k} \sqrt{y} + C \\
T &= -\frac{2A}{k} \sqrt{0} + C &\color{red} y=0,t=T \\
T=C \\
t &= -\frac{2A}{k} \sqrt{y} + T \\
\frac{2A}{k} \sqrt{y} &= T-t \color{red} \cdots (1) \\
\frac{2A}{k} \sqrt{y_0} &= T-0 \color{red} \cdots (2) &\color{red} y=y_0,t=0 \\
\frac{\sqrt{y}}{\sqrt{y_0}} &= \frac{T-t}{T} &\color{red} (1) \div (2) \\
&= 1-\frac{t}{T} \\
\frac{y}{y_0} &= \bigg(1-\frac{t}{T} \bigg)^2 \\
\therefore y &= y_0 \bigg(1-\frac{t}{T} \bigg)^2
\end{aligned} \)

(c)    Suppose it takes \(10\) seconds for half the water to drain out. How long does it take to empty the full cooler?

\( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}
\frac{1}{2}y_0 &= y_0\bigg(1-\frac{10}{T}\bigg)^2 &\color{red} t=10,y= \frac{1}{2} y_0 \\
\frac{1}{2} &= \bigg(1-\frac{10}{T}\bigg)^2 \\
\frac{1}{\sqrt{2}} &= 1-\frac{10}{T} \\
\frac{10}{T} &= 1-\frac{1}{\sqrt{2}} \\
\frac{10}{T} &= \frac{\sqrt{2}-1}{\sqrt{2}} \\
\frac{T}{10} &= \frac{\sqrt{2}}{\sqrt{2}-1} \\
T &= \frac{10\sqrt{2}}{\sqrt{2}-1} \\
\therefore T &\approx 34.1
\end{aligned} \)

 

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume




Your email address will not be published. Required fields are marked *