Respuesta :
It takes 8.35 hours for the size of the sample to double
Step-by-step explanation:
The form of the continuous exponential growth model is
[tex]A=Pe^{rt}[/tex] , where
- A is the new value
- P is the initial value
- r is the rate of growth or decay in decimal
- t is the time
∵ The growth rate is 8.3%
∴ r = 8.3 ÷ 100 = 0.083
∵ The size of the sample will doubled in t hours
∴ A = 2 P
Use the formula of the contentious exponential growth above
∵ [tex]2P=Pe^{0.083t}[/tex]
- Divide both sides by P
∴ [tex]2=e^{0.083t}[/tex]
- Insert ㏑ in both sides
∴ [tex]ln(2)=ln(e^{0.083t})[/tex]
- Remember [tex]ln(e^{n})=n[/tex] because ㏑(e) = 1
∴ [tex]ln(2)=0.083t[/tex]
- Divide both sides by 0.083
∴ [tex]\frac{ln(2)}{0.083}=t[/tex]
∴ t = 8.35 hours to the nearest hundredth
It takes 8.35 hours for the size of the sample to double
Learn more:
You can learn more about the logarithmic function in brainly.com/question/1447265
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After 1.15 hours it takes for the size of the sample to double.
What function represent exponential growth?
An exponential function is represented as:
[tex]\rm y=a(1+r)^x[/tex]
Where
- r represents the growth rate (i.e. 8.3%).
- x represents the number of hours.
- y represents the current population.
- a represents the initial population.
When the population doubles, we have:
y = 2a
So, the equation becomes:
[tex]\rm y=a(1+r)^x\\\\2a=a(1+0.83)^x\\\\2=(1.83)^x[/tex]
Taking log on both sides
[tex]\rm 2=(1.83)^x\\\\log2=log(1.83)^x\\\\ log 2= x log(1.83)\\\\x= \dfrac{log2}{log1.83}\\\\x=1.155[/tex]
Hence, After 1.15 hours it take for the size of the sample to double.
To know more about the Exponential function click the link given below.
https://brainly.com/question/2901380
