MATH SOLVE

4 months ago

Q:
# a culture started with 6000 bacteria . after 5 hours it grew 7800 bacteria. predict how many bacteria will be present after 13 hours . use the formula P = Ae^kt round the nearest whole number

Accepted Solution

A:

We know that we need to use the formula: [tex]P=Ae^{kt}[/tex]

where

[tex]P[/tex] is final population of bacteria after [tex]t[/tex] hours

[tex]A[/tex] is the initial population

[tex]k[/tex] is the growth rate

[tex]t[/tex] is the time in hours

The first thing we are going to do is find [tex]k[/tex]. We know for our problem that: [tex]P=7800[/tex], [tex]A=6000[/tex], and [tex]t=5[/tex]. Lets replace those values in our formula:

[tex]P=Ae^{kt}[/tex]

[tex]7800=6000e^{5k}[/tex]

Now, to find [tex]k[/tex] we are going to isolate [tex]e^{5k}[/tex], and then apply logarithms:

[tex] \frac{7800}{6000} =e^{5k}[/tex]

[tex]e^{5k}= \frac{13}{10} [/tex]

[tex]ln(e^{5k})=ln( \frac{13}{10} )[/tex]

[tex]5k=ln( \frac{13}{10}) [/tex]

[tex]k= \frac{ln( \frac{13}{10}) }{5} [/tex]

[tex]k=0.052[/tex]

Now that we have [tex]k[/tex], we are going to use our formula one more time, but this time [tex]t=13[/tex]:

[tex]P=Ae^{kt}[/tex]

[tex]P=6000e^{(0.052)(13)[/tex]

[tex]P=6000e^{0.676}[/tex]

[tex]P=11796[/tex]

We can conclude that the culture will have 11796 bacteria after 13 hours.

where

[tex]P[/tex] is final population of bacteria after [tex]t[/tex] hours

[tex]A[/tex] is the initial population

[tex]k[/tex] is the growth rate

[tex]t[/tex] is the time in hours

The first thing we are going to do is find [tex]k[/tex]. We know for our problem that: [tex]P=7800[/tex], [tex]A=6000[/tex], and [tex]t=5[/tex]. Lets replace those values in our formula:

[tex]P=Ae^{kt}[/tex]

[tex]7800=6000e^{5k}[/tex]

Now, to find [tex]k[/tex] we are going to isolate [tex]e^{5k}[/tex], and then apply logarithms:

[tex] \frac{7800}{6000} =e^{5k}[/tex]

[tex]e^{5k}= \frac{13}{10} [/tex]

[tex]ln(e^{5k})=ln( \frac{13}{10} )[/tex]

[tex]5k=ln( \frac{13}{10}) [/tex]

[tex]k= \frac{ln( \frac{13}{10}) }{5} [/tex]

[tex]k=0.052[/tex]

Now that we have [tex]k[/tex], we are going to use our formula one more time, but this time [tex]t=13[/tex]:

[tex]P=Ae^{kt}[/tex]

[tex]P=6000e^{(0.052)(13)[/tex]

[tex]P=6000e^{0.676}[/tex]

[tex]P=11796[/tex]

We can conclude that the culture will have 11796 bacteria after 13 hours.