08Bconstantrates

= Constant Rates of Change =


 * If the rate at which a quantity is changing does not alter,
 * we call it a ** constant rate **.


 * For example
 * If water flows into a water tank at 1.2 litres/minute
 * the volume of water in the tank is increasing at a constant rate


 * A graph of the volume of water in the tank (see right)
 * shows a __straight line__ where
 * the gradient of the graph = 1.2
 * the __gradient__ of the graph represents the __rate__ the tank is filling


 * If the rate was not constant,
 * a graph of the situation would __not__ produce a straight line


 * Like gradients, a rate of change can be
 * ** positive ** if the quantity is increasing
 * the graph slopes up
 * ** zero ** if the quantity is not changing
 * the graph is horizontal
 * ** negative ** if the quantity is decreasing
 * the graph slopes down


 * Example 1 **

The height (h) of a plant (in cm) was measured every week from the time it was planted. The results are shown on the graph here:


 * 1) Find the gradient of the graph
 * 2) find the rate at which the plant is growing
 * 3) find the rule for the height of the plant as a function of time (in weeks)


 * Solution:**


 * ** The graph shows a straight line so the rate is constant (and positive) **


 * ** select any two points on the graph **
 * I have chosen: (2, 6) and (4, 9)


 * ** Now calculate gradient **

math . \qquad m = \dfrac{y_2 - y_1}{x_2-x_1} \\.\\ . \qquad m = \dfrac{9-6}{4-2} \\.\\ . \qquad m = \dfrac{3}{2} \\.\\ . \qquad m = 1.5 math


 * ** Hence the plant is growing at a rate of 1.5 cm/week **


 * ** Observe the y-intercept from the graph **
 * c =3


 * ** Hence the rule for the graph is **
 * h = 1.5w + 3

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