Sure, let's go through each part of the question step-by-step:
### (a) Relative Rate of Growth
The relative rate of growth of a population described by the exponential function [tex]\( n(t) = 950 e^{0.35 t} \)[/tex] can be determined by examining the exponent of the base [tex]\( e \)[/tex].
In this function, the exponent is [tex]\( 0.35 t \)[/tex]. The coefficient of [tex]\( t \)[/tex] in the exponent, [tex]\( 0.35 \)[/tex], represents the relative rate of growth.
So, the relative rate of growth is [tex]\( 0.35 \)[/tex] or [tex]\( 35\% \)[/tex] per hour.
Your answer is [tex]\( 35 \)[/tex] percent.
### (b) Initial Population of the Culture
The initial population is found by evaluating the function [tex]\( n(t) \)[/tex] at [tex]\( t = 0 \)[/tex].
[tex]\[ n(0) = 950 e^{0.35 \cdot 0} \][/tex]
Since [tex]\( e^0 = 1 \)[/tex]:
[tex]\[ n(0) = 950 \times 1 = 950 \][/tex]
So, the initial population of the culture is [tex]\( 950 \)[/tex].
Your answer is [tex]\( 950 \)[/tex].
### (c) Population at [tex]\( t = 5 \)[/tex]
To find the population at [tex]\( t = 5 \)[/tex], we substitute [tex]\( t = 5 \)[/tex] into the function [tex]\( n(t) \)[/tex]:
[tex]\[ n(5) = 950 e^{0.35 \cdot 5} \][/tex]
Evaluating the exponent:
[tex]\[ 0.35 \times 5 = 1.75 \][/tex]
So, we have:
[tex]\[ n(5) = 950 e^{1.75} \][/tex]
Using the exponential value:
[tex]\[ n(5) \approx 950 \times 5.46687254 \][/tex]
[tex]\[ n(5) \approx 5466.87254 \][/tex]
Rounding it to a reasonable precision, we can say that the population at [tex]\( t = 5 \)[/tex] is approximately [tex]\( 5466.872542205444 \)[/tex].
Your answer is [tex]\( 5466.872542205444 \)[/tex].
Hope this helps!
