Join the IDNLearn.com community and start getting the answers you need today. Find the information you need quickly and easily with our comprehensive and accurate Q&A platform.
Sagot :
Alright, let's solve each part of this problem step-by-step.
### Part (a) - Relative Rate of Growth
The given function for the number of bacteria is:
[tex]\[ n(t) = 925 e^{0.1 t} \][/tex]
The relative rate of growth in an exponential function of the form [tex]\( n(t) = n_0 e^{rt} \)[/tex] is given by the exponent [tex]\( r \)[/tex].
Here, the exponent [tex]\( r \)[/tex] is [tex]\( 0.1 \)[/tex].
To express the relative rate of growth as a percentage, we multiply by 100:
[tex]\[ 0.1 \times 100 = 10\% \][/tex]
So, the relative rate of growth of this bacterium population is [tex]\( 10\% \)[/tex].
### Part (b) - Initial Population at [tex]\( t = 0 \)[/tex]
To find the initial population, we evaluate the function [tex]\( n(t) \)[/tex] at [tex]\( t = 0 \)[/tex]:
[tex]\[ n(0) = 925 e^{0.1 \cdot 0} \][/tex]
Since [tex]\( e^{0} = 1 \)[/tex]:
[tex]\[ n(0) = 925 \times 1 = 925 \][/tex]
Therefore, the initial population of the culture is [tex]\( 925 \)[/tex].
### Part (c) - Population at [tex]\( t = 5 \)[/tex]
To find the population at [tex]\( t = 5 \)[/tex], we evaluate the function [tex]\( n(t) \)[/tex] at [tex]\( t = 5 \)[/tex]:
[tex]\[ n(5) = 925 e^{0.1 \cdot 5} \][/tex]
First, calculate the exponent:
[tex]\[ 0.1 \times 5 = 0.5 \][/tex]
So, we need to find [tex]\( e^{0.5} \)[/tex], and then multiply by 925.
Evaluating:
[tex]\[ n(5) = 925 e^{0.5} \][/tex]
The approximate value of [tex]\( e^{0.5} \)[/tex] is used to find:
[tex]\[ n(5) \approx 925 \times \text{value of } e^{0.5} \][/tex]
After evaluating, you would get approximately:
[tex]\[ n(5) \approx 1525.0671753976185 \][/tex]
Therefore, the population at [tex]\( t = 5 \)[/tex] is approximately [tex]\( 1525.0671753976185 \)[/tex].
So, here are the answers:
(a) The relative rate of growth is [tex]\( 10\% \)[/tex].
(b) The initial population of the culture is [tex]\( 925 \)[/tex].
(c) The population at [tex]\( t = 5 \)[/tex] is approximately [tex]\( 1525.0671753976185 \)[/tex].
### Part (a) - Relative Rate of Growth
The given function for the number of bacteria is:
[tex]\[ n(t) = 925 e^{0.1 t} \][/tex]
The relative rate of growth in an exponential function of the form [tex]\( n(t) = n_0 e^{rt} \)[/tex] is given by the exponent [tex]\( r \)[/tex].
Here, the exponent [tex]\( r \)[/tex] is [tex]\( 0.1 \)[/tex].
To express the relative rate of growth as a percentage, we multiply by 100:
[tex]\[ 0.1 \times 100 = 10\% \][/tex]
So, the relative rate of growth of this bacterium population is [tex]\( 10\% \)[/tex].
### Part (b) - Initial Population at [tex]\( t = 0 \)[/tex]
To find the initial population, we evaluate the function [tex]\( n(t) \)[/tex] at [tex]\( t = 0 \)[/tex]:
[tex]\[ n(0) = 925 e^{0.1 \cdot 0} \][/tex]
Since [tex]\( e^{0} = 1 \)[/tex]:
[tex]\[ n(0) = 925 \times 1 = 925 \][/tex]
Therefore, the initial population of the culture is [tex]\( 925 \)[/tex].
### Part (c) - Population at [tex]\( t = 5 \)[/tex]
To find the population at [tex]\( t = 5 \)[/tex], we evaluate the function [tex]\( n(t) \)[/tex] at [tex]\( t = 5 \)[/tex]:
[tex]\[ n(5) = 925 e^{0.1 \cdot 5} \][/tex]
First, calculate the exponent:
[tex]\[ 0.1 \times 5 = 0.5 \][/tex]
So, we need to find [tex]\( e^{0.5} \)[/tex], and then multiply by 925.
Evaluating:
[tex]\[ n(5) = 925 e^{0.5} \][/tex]
The approximate value of [tex]\( e^{0.5} \)[/tex] is used to find:
[tex]\[ n(5) \approx 925 \times \text{value of } e^{0.5} \][/tex]
After evaluating, you would get approximately:
[tex]\[ n(5) \approx 1525.0671753976185 \][/tex]
Therefore, the population at [tex]\( t = 5 \)[/tex] is approximately [tex]\( 1525.0671753976185 \)[/tex].
So, here are the answers:
(a) The relative rate of growth is [tex]\( 10\% \)[/tex].
(b) The initial population of the culture is [tex]\( 925 \)[/tex].
(c) The population at [tex]\( t = 5 \)[/tex] is approximately [tex]\( 1525.0671753976185 \)[/tex].
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. IDNLearn.com provides the best answers to your questions. Thank you for visiting, and come back soon for more helpful information.