Answer:
a. ![y=6(1.7472)^x](https://tex.z-dn.net/?f=y%3D6%281.7472%29%5Ex)
b. ![y=6e^{0.558t}](https://tex.z-dn.net/?f=y%3D6e%5E%7B0.558t%7D)
c.13.3 months
Step-by-step explanation:
a.-Given the first term at
is 6 and the second term at
is 32.
-Let's take rabbit population as a function of time to be
![y=ab^x](https://tex.z-dn.net/?f=y%3Dab%5Ex)
where y is the population at time x and a the initial population at ![t_0\\](https://tex.z-dn.net/?f=t_0%5C%5C)
#We substitute our values to calculate the value of the constant b:
![y_x=ab^x\\\\y_3=ab^3\\\\32=6b^3\\\\b=1.472](https://tex.z-dn.net/?f=y_x%3Dab%5Ex%5C%5C%5C%5Cy_3%3Dab%5E3%5C%5C%5C%5C32%3D6b%5E3%5C%5C%5C%5Cb%3D1.472)
#Replace b in the population function:
![y=ab^x, b=1.7472,a=6\\\\\therefore y=6(1.7472)^x](https://tex.z-dn.net/?f=y%3Dab%5Ex%2C%20b%3D1.7472%2Ca%3D6%5C%5C%5C%5C%5Ctherefore%20y%3D6%281.7472%29%5Ex)
Hence, the regression for the rabbit population as a function of time x is ![y=6(1.7472)^x](https://tex.z-dn.net/?f=y%3D6%281.7472%29%5Ex)
b. The exponential function in terms of base
is usually expressed as:
![A=A_0e^{kt}](https://tex.z-dn.net/?f=A%3DA_0e%5E%7Bkt%7D)
Where:
-is the initial population at ![t_o](https://tex.z-dn.net/?f=t_o)
-is the population at time t.
is the exponential growth constant.
the exponent
Our function in terms of base exponent is rewritten as:
![y=A_0e^{kt}](https://tex.z-dn.net/?f=y%3DA_0e%5E%7Bkt%7D)
#Substitute with actual figures to solve for t:
![y=A_0e^{kt}, y=32, xt=3, A_0=6\\\\32=6e^{3k}\\\\3k=In (32/6)\\\\k=0.5580](https://tex.z-dn.net/?f=y%3DA_0e%5E%7Bkt%7D%2C%20y%3D32%2C%20xt%3D3%2C%20A_0%3D6%5C%5C%5C%5C32%3D6e%5E%7B3k%7D%5C%5C%5C%5C3k%3DIn%20%2832%2F6%29%5C%5C%5C%5Ck%3D0.5580)
Hence, the regression equation in terms of base e is ![y=6e^{0.558t}](https://tex.z-dn.net/?f=y%3D6e%5E%7B0.558t%7D)
c. We substitute y with any number higher than 10,000 to estimate the time for the rabbits to exceed 10,000.
-We know that ![y=6e^{0.558t}.](https://tex.z-dn.net/?f=y%3D6e%5E%7B0.558t%7D.)
Therefore we calculate t as(take y=10001):
![y=6e^{0.558t}, y=10001\\\\10001=6e^{0.558t}\\\\1666.8333=e^{0.558t}\\\\0.558t= In 1666.8333\\\\t=13.2951](https://tex.z-dn.net/?f=y%3D6e%5E%7B0.558t%7D%2C%20y%3D10001%5C%5C%5C%5C10001%3D6e%5E%7B0.558t%7D%5C%5C%5C%5C1666.8333%3De%5E%7B0.558t%7D%5C%5C%5C%5C0.558t%3D%20In%201666.8333%5C%5C%5C%5Ct%3D13.2951)
Hence, it takes approximately 13.3 months for the population to exceed 10000