Question

A population of mold decays at a rate of 20 mold spores per day. Assume that the initial population of mold is 980 spores. How many days will it take for the population to be less than 250 mold spores? Round your answer up to the nearest whole number.

Answer 1

In this problem, we have an exponential decay function of the form

$y=a(b^x)$

where

b is less than 1

b<1

we have

a=980 spores (initial value)

so

$y=980(b^x)$

we know that

the population of mold decays at a rate of 20 mold spores per day

so

x=1 -----> y=980-20=960

x=2 -----> y=960-20=940

For x=1, y=960

substitute in the equation

$960=980(b^1)$

Solve for b

b=960/980

simplify

b=48/49

therefore

the equation is

$y=980(\frac{48}{49})^x$

For y=250 mold spores

substitute

$250=980(\frac{48}{49})^x$

Solve for x

$\frac{250}{980}=(\frac{48}{49})^x$

Apply log both sides

$\log (\frac{250}{980})=x\cdot\log (\frac{48}{49})$

x=66 days

therefore

The number of days must be greater than 66 days

so

The answer is

67 days