Linear Growth

Linear growth or linear decay is present when the change of a value $N$ , with the same change over time, is constant.

In other words, the quantity changes by the same amount within equal time intervals. The linear growth function is given by a line equation:

$\begin{array}{l} N (t) = a \cdot t + N_{0} \end{array}$

Here we have:

$N (t)$ : the number or amount of $N$ after time $t$ ,
$a$ : the rate of change,
$N_{0}$ : the number or amount of $N$ at time $0$ , i..e the initial value

Properties

The growth rate or rate of change $a$ is constant for linear growth or decay: $a \in ℝ$ .
It corresponds to the slope of the graph of the linear growth function.
Monotonicity: If $a > 0$ , we speak of linear growth. The function is then strictly monotonically increasing.
If $a < 0$ , the function describes linear decay. The function is then strictly monotonically decreasing.

The graph of a linear growth function

As with linear functions, the rate of change $a$ is calculated using a gradient triangle.

$Δ N (t)$ denotes the difference of the values of $N$ at two points in time.

In the figure, this is:

Δ N (t) = N (5) - N (0)

$Δ t$ stands for the period of time during which $N$ is observed. In the figure:

Δ t = 5 - 0 = 5

Example

A tree is planted in a garden. At this point it protrudes $1 m$ from the ground. After how many years will the tree be $5 m$ high if it grows by $10 c m$ a year on average?

Solution:

First, write down the given and searched values from the data.

Required is the time $t$ at which the tree has reached the size $5 m$ .

Given is the size of the tree at the beginning (= starting value $N_{0}$ ), its growth rate (= rate of change $a$ ) and its size reached after $t$ years (= $N (t)$ ).

(Note: t is given in years, $N$ is the size of the tree in metres.

The tree grows $10 c m$ per year, therefore the unit of $a : \frac{c m}{y e a r}$ ).

$N_{0} = 1 m$

$a = 10 \frac{c m}{year} = 0.1 \frac{m}{year}$

$N (t) = 5 m$

Now substitute the given values into the functional equation $N (t) = a \cdot t + N_{0}$ and solve the equation for the $t$ we are looking for.

$\begin{array}{rcll} 5 m & = & 0.1 \frac{m}{y e a r} \cdot t + 1 m & | - 1 m \\ 4 m & = & 0.1 \frac{m}{y e a r} \cdot t & | : 0.1 \frac{m}{y e a r} \\ \frac{4 m}{0.1 \frac{m}{y e a r}} & = & t \\ 40 y e a r (s) & = & t \end{array}$

Answer: After 40 years the tree is $5 m$ high.

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