Respuesta :
Answer:
3.3 hours.
Step-by-step explanation:
We have been given that a scientist is testing a new antibiotic by applying the antibiotic to a colony of 10,000 bacteria. The number of bacteria decreases by 75% every two hours.
Since number of bacteria is decreasing exponentially, so we will use exponential decay function.
[tex]y=a*b^x[/tex], where,
a= Initial value,
b = For decay b is in form (1-r), where r is rate in decimal form.
Let us convert our given rate in decimal form.
[tex]75\%=\frac{75}{100}=0.75[/tex]
As number of bacteria is decreasing every 2 hours, so number of bacteria decreased in 1 hour will be x/2.
Upon substituting our given values in above formula we will get,
[tex]y=10,000(1-0.75)^{\frac{x}{2}}[/tex]
To find the number of hours it will take to for the bacteria colony to decrease to 1000, we will substitute y = 1,000 in our equation.
[tex]1,000=10,000(0.25)^{\frac{x}{2}}[/tex]
Let us divide both sides of our equation by 10,000.
[tex]\frac{1,000}{10,000}=\frac{10,000(0.25)^{\frac{x}{2}}}{10,000}[/tex]
[tex]0.1=(0.25)^{\frac{x}{2}}[/tex]
Let us take natural log of both sides of our equation.
[tex]ln(0.1)=ln((0.25)^{\frac{x}{2}})[/tex]
[tex]ln(0.1)=\frac{x}{2}*ln(0.25)[/tex]
[tex]-2.302585=\frac{x}{2}*-1.386294[/tex]
[tex]x=\frac{-2.302585}{-1.386294}*2[/tex]
[tex]x=1.660964*2[/tex]
[tex]x=3.3219\approx 3.3[/tex]
Therefore, it will take 3.3 hours for the bacteria colony to decrease to 1000.
