Let's break down the solution step by step.
(a) Exponential Rate of Growth:
The number of bacteria in the culture is given by the function:
[tex]\[ n(t) = 995 e^{0.35 t} \][/tex]
In this function, the term inside the exponential, [tex]\( 0.35 \)[/tex], represents the growth rate of the bacteria. This rate is in decimal form. To convert it to a percentage, we multiply by 100:
[tex]\[ 0.35 \times 100 = 35 \% \][/tex]
So, the exponential rate of growth of this bacterium population is 35%.
Answer: 35%
(b) Initial Population at [tex]\( t = 0 \)[/tex]:
To find the initial population of the culture, we substitute [tex]\( t = 0 \)[/tex] into the given function:
[tex]\[ n(0) = 995 e^{0.35 \times 0} \][/tex]
Simplifying the exponent:
[tex]\[ n(0) = 995 e^0 \][/tex]
Since [tex]\( e^0 = 1 \)[/tex], we have:
[tex]\[ n(0) = 995 \times 1 = 995 \][/tex]
So, the initial population of the culture at [tex]\( t = 0 \)[/tex] is 995.
Answer: 995
(c) Population at [tex]\( t = 4 \)[/tex]:
To find the population at [tex]\( t = 4 \)[/tex], we substitute [tex]\( t = 4 \)[/tex] into the given function:
[tex]\[ n(4) = 995 e^{0.35 \times 4} \][/tex]
Calculating the exponent:
[tex]\[ 0.35 \times 4 = 1.4 \][/tex]
So,
[tex]\[ n(4) = 995 e^{1.4} \][/tex]
Given that:
[tex]\[ e^{1.4} \approx 4.034923967010451 \][/tex]
We multiply this by the initial population:
[tex]\[ n(4) = 995 \times 4.034923967010451 \approx 4034.923967010451 \][/tex]
So, the population at [tex]\( t = 4 \)[/tex] is approximately 4034.923967010451.
Answer: 4034.923967010451
