Question

in cell reproduction, there are 2 cells present after day 1; 4 after day 2, 8 after day 3 and so on. write an exponential equation that could be used to find out how many cells there will be on the nth day? how many cells will there be on the 20th day?

Explanation:

Step1: Determine the general form of the exponential equation

The general form of an exponential equation is \(y = a\cdot b^{x}\), where \(a\) is the initial value and \(b\) is the base.
When \(x = 1\) (day 1), \(y=2\). When \(x = 2\) (day 2), \(y = 4\). Substituting \(x = 1,y = 2\) into \(y=a\cdot b^{x}\), we get \(2=a\cdot b^{1}\). Substituting \(x = 2,y = 4\) into \(y=a\cdot b^{x}\), we get \(4=a\cdot b^{2}\).
Dividing the second equation by the first equation: \(\frac{4}{2}=\frac{a\cdot b^{2}}{a\cdot b^{1}}\), which simplifies to \(b = 2\).
Substituting \(b = 2\) into \(2=a\cdot b^{1}\), we get \(a = 1\). So the exponential equation is \(y=2^{x}\).

Step2: Calculate the number of cells on the 20th day

When \(x = 20\), substitute into \(y = 2^{x}\). So \(y=2^{20}\).
Using the formula \(a^{m}\) where \(a = 2\) and \(m=20\), \(2^{20}=1048576\).

Answer:

The exponential equation is \(y = 2^{x}\) and the number of cells on the 20th day is \(1048576\).