The population of a Midwestern city decays exponentially. If the population decreased from 900,000 to 800,000 from 2003 to 2005, what will be the population in 2008?
Please help I got 670,422 and the computer marked it round i then tried rounding it to 671000 but it was still wrong! Please help me THank you
computer marked it wrong*
Its still wrong omg I dont know why
the full solution is in the related questions at the bottom of the page
can someone explain step by step how to do this
To find the population in 2008, we need to determine the rate at which the population is decaying. The exponential decay formula can be used to represent this situation:
P(t) = P₀ * e^(-kt)
Where:
P(t) is the population at time t
P₀ is the initial population
k is the decay constant
t is the time period
To find the decay constant (k), we can use the given information about the population decreasing from 900,000 to 800,000 from 2003 to 2005:
P(t₁) = P₀ * e^(-k * t₁)
P(t₂) = P₀ * e^(-k * t₂)
Given:
P(t₁) = 800,000
P(t₂) = 900,000
t₁ = 2005 - 2003 = 2 (time period from 2003 to 2005)
Substituting the known values:
800,000 = P₀ * e^(-2k)
900,000 = P₀ * e^(-k)
Dividing the second equation by the first equation, we can eliminate P₀:
900,000 / 800,000 = e^(-k) / e^(-2k)
Simplifying:
1.125 = e^k / e^(2k)
1.125 = 1 / e^k (since e^(2k) / e^k = e^(2k - k) = e^k)
Taking the natural logarithm (ln) of both sides:
ln(1.125) = ln(1 / e^k)
ln(1.125) = -k
Now we can solve for k:
k = -ln(1.125)
Using a calculator, we find that k ≈ 0.1178.
Now that we have the decay constant (k), we can use it to find the population at any given time. In this case, we want to find the population in 2008, which is 3 years after 2005:
t = 2008 - 2005 = 3
Plugging in the values:
P(3) = P₀ * e^(-k * 3)
Substituting the known value for P₀ (800,000):
P(3) = 800,000 * e^(-0.1178 * 3)
Using a calculator, we find that P(3) ≈ 706,237.
Therefore, the population in 2008 should be approximately 706,237.
the two given population numbers only have one significant digit
either 670000 or 700000 is what the computer is looking for
900000e^5(1/2ln(1/8)
This is ithe answer if you were wondering lmao who would have known