Question

The following data were obtained from an experiment on 2 plant samples showing exponential growth: $\begin{array}{cccc}\text{Sample}& \text{Initial weight}& \text{Final weight (in grams)}& \text{Time of growth(in}\\ & \text{grams)}& & \text{seconds)}\\ \text{A}& 107& 321& 15\\ \text{B}& 723& 1446& 16\end{array}$
Can you determine which sample is growing faster?

• A. A
• B. B
• C. They are growing at equal rates
• D. Cannot be determined

Answer 1

The correct option is A A

Sample A	Sample B
Given: Initial weight, W0 = 107 g Final weight, W1 = 321 g Time of growth, t = 15 s	Given: Initial weight, W0 = 723 g Final weight, W1 = 1446 g Time of growth, t = 16 s
To find: Relative growth rate, r	To find: Relative growth rate, r
For exponential growth, $W_1=W_0{e}^{rt}$ ...equation (i) where $W_1$ is the final size of a sample, $W_0$ is the initial size of the sample, r is the relative growth rate, t is the time of growth and e is the base of natural logarithms. $\therefore 321=107\times e^{r\times 15}\Rightarrow 3=e^{r\times 15}\Rightarrow lo{g}_{e}3=15r\Rightarrow r=\frac{lo{g}_{e}3}{15}\Rightarrow r=\frac{1.099}{15}\Rightarrow r=0.073s^{-1}$	For exponential growth, $W_1=W_0{e}^{rt}$ ...equation (i) where $W_1$ is the final size of a sample, $W_0$ is the initial size of the sample, r is the relative growth rate, t is the time of growth and e is the base of natural logarithms. $\therefore 1446=723\times e^{r\times 16}\Rightarrow 2=e^{r\times 16}\Rightarrow lo{g}_{e}2=16r\Rightarrow r=\frac{lo{g}_{e}2}{16}\Rightarrow r=\frac{0.693}{16}\Rightarrow r=0.043s^{-1}$

Since the relative growth rate of sample A is higher than that of sample B, sample A is growing faster than sample B.