Question

A population of mold decays at a rate of 25 mold spores per day. Assume that the initial population of mold spores is 1640 spores.Step 2 of 2 : How many days will it take for the population to be less than 350 mold spores? Round your answer up to the nearest whole number.

Answer 1

The rate of decay of the mold population is 25 mold spores per day or 25%.

The initial mold population is assumed to be 1640 spores.

The formula for exponential decay is;

$f(x)=a(1-r)^x$

Where the variables are;

$\begin{gathered} r=0.25 \\ a=1640 \\ x=Number\text{ of days} \end{gathered}$

We can substitute into the formula and we'll have;

$f(x)=1640(1-0.25)^x$

For the population of spores to be 350 mold spores, the equation would now become;

$\begin{gathered} 350=1640(1-0.25)^x \\ 350=1640(0.75)^x \\ \text{Divide both sides by 1640;} \\ \frac{350}{1640}=\frac{1640(0.75)^x}{1640} \\ 0.2134=0.75^x \end{gathered}$

At this point we shall apply the exponent rule;

$\begin{gathered} f(x)=g(x) \\ \text{Then,} \\ \ln (f(x))\ln (g(x)) \end{gathered}$

Hence;

$\ln (0.2134)=\ln (0.75^x)^{}$

Next we apply the log rule;

$\log _bx^a=a\log _bx$

We now re-write our equation as;

$\begin{gathered} \ln (0.2134)=x\ln (0.75) \\ x=\frac{\ln (0.2134)}{\ln (0.75)} \end{gathered}$

By use of a calculator, the value of x now becomes;

$\begin{gathered} x=5.36907 \\ \text{Rounded to the nearest whole number,} \\ x=5 \end{gathered}$

That means it would take approximately 5 days for the mold population to reach 350 spores.

Hence, for the population to be less than 350 would take a