Answer :
Answer:
The value of the quantity after 21 days is 2,347.59.
Step-by-step explanation:
The exponential growth function is
[tex]A=A_0(1+r)^t[/tex]
A= The number of quantity after t days
[tex]A_0[/tex]= initial number of quantity
r= rate of growth
t= time in days.
A quantity with an initial value of 830 grows at a rate such that the quantity doubles in 2 weeks = 14 days.
Now A= (2×830)= 1660
[tex]A_0[/tex] = 830
t = 14 days
r=?
Now plug all value in exponential growth function
[tex]1660=830(1+r)^{14}[/tex]
[tex]\Rightarrow \frac{1660}{830}= (1+r)^{14}[/tex]
[tex]\Rightarrow 2= (1+r)^{14}[/tex]
[tex]\Rightarrow (1+r) ^{14}=2[/tex]
[tex]\Rightarrow (1+r)=\sqrt[14]{2}[/tex]
[tex]\Rightarrow r=\sqrt[14]{2}-1[/tex]
Now, to find the quantity after 21 days, we plug [tex]A_0[/tex] = 830, t= 21 days in exponential function
[tex]A=830( 1+\sqrt[14]{2}-1)^{21}[/tex]
[tex]\Rightarrow A=830(\sqrt[14]2)^{21}[/tex]
[tex]\Rightarrow A=830(2)^\frac{21}{14}[/tex]
[tex]\Rightarrow A=830(2)^\frac{3}{2}[/tex]
[tex]\Rightarrow A=2,347.59[/tex]
The value of the quantity after 21 days is 2,347.59.