08Eratesof+change


 * Rates of change **

At the beginning of chapter five, we covered the theory related to average rates of change and intantaneous rates of change.


 * Example 1 **

The pressure P (Pa), of a given mass kept at constant temperature, and its volume V (m 3 ) are connected by the formula PV = 500 (joules). ... ... (a) Find the rate of change of the pressure with respect to the volume. ... ... (b) What is the rate of change when V = 10m 3.


 * __Solution:__**

(a) Find the rate of change of the pressure with respect to the volume.

math . \qquad PV = 500 \qquad. \\ . \\ .\qquad\;P=\dfrac{500}{V} \qquad. \\ . \\ . \qquad \; P = 500V^{-1} \qquad math

math .\qquad\dfrac{dP}{dV}=-500V^{-2} \qquad. \\ . \\ .\qquad\dfrac{dP}{dV}=\dfrac{-500}{V^2} math

The rate of change is negative since the pressure is decreasing as the volume increases.

(b) What is the rate of change when V = 10m 3.

math . \qquad \text{When V=10}\qquad\;\dfrac{dP}{dV}=\dfrac{-500}{10^2} \qquad. math

math .\qquad\dfrac{dP}{dV}=-5 \; Pa/m^3 \qquad. math

The rate of change of the pressure when V = 10m 3 is –5 Pa/m 3 (a decreasing rate of 5 Pa/m 3 ).

An understanding of chemistry etc is not required to take the units given in the question and put them into your answer.
 * NOTE:** The units of dP/dV come from the units of P (Pa) divided by the units of V (m 3 )


 * Example 2 **

A water tank is being emptied and the quantity of water, Q litres, remaining in the tank t minutes after it starts to empty is given by math . \qquad Q(t)=1000(20-t)^2, \qquad t \geqslant 0 \qquad. math

... ... (i) At what rate is the tank being emptied at any time t? .... ... (ii) How long does it take to empty the tank? ... ... (iii) At what time is the water flowing out at the rate of 20,000 lites per minute? ... ... (iv) What is the average rate at which the water flows out in the first 5 minutes.


 * Note:** In modelling questions, time will always be greater than or equal to zero. We do not consider negative time.


 * Note:** If asked to graph this, only graph the first quadrant. Neither t nor Q can become negative so reflect that in your graph.


 * __Solution:__**

(i) At what rate is the tank being emptied at any time t?

math . \qquad Q(t)=1000(20-t)^2 \qquad. \\ . \\ . \qquad \dfrac{dQ}{dt}=-2000(20-t) \; \text{litres/minute} \qquad. math


 * Note**: The units for dQ/dt come from the units of Q (litres) divided by the units for t (minutes).

(ii) How long does it take to empty the tank?

... ... Tank is empty when Q = 0

... ... Q(t) = 0

... ... 1000 (20 – t) 2 = 0

... ... t = 20 minutes

(iii) At what time is the water flowing out at the rate of 20,000 lites per minute?

math . \qquad \text{Find } t \text{ when } \dfrac{dQ}{dt}=-20000\quad \{ \textit{negative since water is flowing out} \} \qquad. math

math \\ . \qquad \dfrac{dQ}{dt}=-2000(20-t) \qquad. \\ . \\ . \qquad -2000 (20 - t) = -20000 \qquad. \\ . \\ . \qquad 20 - t = 10 \\ .\\ . \qquad t = 10 \;\; \text{ minutes} math

The water flows out at a rate of 20,000 litres per minute exactly 10 minutes after the tank begins to empty.

(iv) What is the average rate at which the water flows out in the first 5 minutes.

math . \qquad \text{Average Rate of Change} \; = \dfrac{ Q(t_2) - Q(t_1) }{ t_2 - t_1} \qquad. math

... ... In the first five minutes so between t = 0 and t = 5

math \\ . \qquad \qquad =\dfrac{Q(5)-Q(0)}{5} \qquad. \\ . \\ . \qquad \qquad =\dfrac{1000(20-5)^2-1000(20-0)^2}{5} \qquad. \\ . \\ . \qquad \qquad=-35000 \;\; \text{ litres per minute} \qquad. math

The average rate of water flow in the first 5 minutes is –35,000 litres/min (a decrease of 35,000 litre per minute).

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