The population of Australia in x years after 1980 can be modeled by the function y=14.6*(1.014)^x. Estimate the population of Australia for each year.
a) 1976
b) 1980
c) 1972
I am not sure of what to do. Help Please ?
in 1976, x = -4
y = 14.6 /1.014^4
in 1980, x = 0
anything^0 =1
y = 14.6
in 1972, x = -8
y = 14.6 /1.014^8
To estimate the population of Australia for each year, you need to substitute the given years into the function and calculate the values.
a) 1976:
To estimate the population of Australia in 1976, we need to find the value of y when x is -4 years (1976 - 1980 = -4). Substituting x = -4 into the formula:
y = 14.6 * (1.014)^x
y = 14.6 * (1.014)^(-4)
Using a calculator, evaluate (1.014)^(-4):
(1.014)^(-4) ≈ 0.986
Substituting this value into the formula:
y = 14.6 * 0.986
y ≈ 14.4
So, the estimated population of Australia in 1976 is approximately 14.4 million.
b) 1980:
To estimate the population of Australia in 1980, we need to find the value of y when x is 0. Substituting x = 0 into the formula:
y = 14.6 * (1.014)^x
y = 14.6 * (1.014)^0
Since any number raised to the power of 0 is 1, the value of y is:
y = 14.6 * 1
y = 14.6
So, the estimated population of Australia in 1980 is 14.6 million.
c) 1972:
To estimate the population of Australia in 1972, we need to find the value of y when x is -8 years (1972 - 1980 = -8). Substituting x = -8 into the formula:
y = 14.6 * (1.014)^x
y = 14.6 * (1.014)^(-8)
Using a calculator, evaluate (1.014)^(-8):
(1.014)^(-8) ≈ 0.971
Substituting this value into the formula:
y = 14.6 * 0.971
y ≈ 14.2
So, the estimated population of Australia in 1972 is approximately 14.2 million.
To estimate the population of Australia for each year, you will substitute the given year into the given function and calculate the value of y.
For option a) 1976:
To estimate the population in 1976, we need to find the value of y when x is -4 (1976 - 1980 = -4).
Substituting x = -4 into the function:
y = 14.6 * (1.014)^(-4)
To calculate this, we need to evaluate the exponent first: (1.014)^(-4) = 0.986307.
Now we can substitute this value back into the equation:
y = 14.6 * 0.986307 = 14.4041
Therefore, the estimated population of Australia in 1976 is approximately 14.4 million.
For option b) 1980:
To estimate the population in 1980, we need to find the value of y when x is 0.
Substituting x = 0 into the function:
y = 14.6 * (1.014)^0
Since any number raised to the power of zero is 1, we can simplify:
y = 14.6 * 1 = 14.6
Therefore, the estimated population of Australia in 1980 is approximately 14.6 million.
For option c) 1972:
To estimate the population in 1972, we need to find the value of y when x is -8 (1972 - 1980 = -8).
Substituting x = -8 into the function:
y = 14.6 * (1.014)^(-8)
Following the same process as before, we find:
(1.014)^(-8) ≈ 0.973783
Substituting this value back into the equation:
y = 14.6 * 0.973783 = 14.1913
Therefore, the estimated population of Australia in 1972 is approximately 14.2 million.