the population of a town is declining at a rate of 4.1% per year. the current population is 25000. 1)what will the population be in 7 years. 2) how long will it take the population to reach 15000?
p = po (.959)^n
p = 25,000 (.959)^7
= 18,650
15,000 = 25, 000 (.959)^n
.6 = .959^n
ln .6 = n ln .959 or any base log
n = 12.2 years
just follow the steps above
To calculate the population in the future and the time it takes for the population to reach a specific value, we need to use a formula that includes exponential decay.
1) To find the population in 7 years, we can use the formula:
Population after t years = Initial population * (1 - r)^t,
where t is the number of years, r is the rate of decline expressed as a decimal, and the initial population is 25000.
Given:
Initial population (P) = 25000
Rate of decline (r) = 4.1% = 0.041 (expressed as a decimal)
Time (t) = 7 years
Using the formula, we can calculate the population after 7 years:
Population after 7 years = 25000 * (1 - 0.041)^7
Population after 7 years ≈ 25000 * (0.959)^7
Population after 7 years ≈ 25000 * (0.808)
Population after 7 years ≈ 20,200 (rounded to the nearest whole number)
Therefore, the population after 7 years will be approximately 20,200.
2) To find out how long it will take for the population to reach 15000, we need to solve the exponential decay formula for t.
Population after t years = P * (1 - r)^t
Given:
Initial population (P) = 25000
Rate of decline (r) = 4.1% = 0.041 (expressed as a decimal)
Target population = 15000
We can set up the equation:
15000 = 25000 * (1 - 0.041)^t
Divide both sides by 25000:
15000 / 25000 = (1 - 0.041)^t
Simplify:
0.6 = 0.959^t
To solve for t, we need to take the logarithm of both sides of the equation. Let's use the natural logarithm (ln):
ln(0.6) = ln(0.959^t)
Using the logarithmic property, we can bring down the exponent:
ln(0.6) = t * ln(0.959)
Divide both sides by ln(0.959):
t = ln(0.6) / ln(0.959)
t ≈ 10.06
Therefore, it will take approximately 10.06 years for the population to reach 15000.
To calculate the population in the future or determine how long it takes for the population to reach a certain level, we can use the formula for exponential growth or decay. In this case, since the population is declining, we will use the formula for exponential decay:
Population = Initial Population × (1 - Rate)^Time
Now let's solve the problems step by step.
1) What will the population be in 7 years?
Given:
Initial Population (P0) = 25000
Rate of decline (r) = 4.1% = 0.041
Time (t) = 7 years
Using the formula for exponential decay:
Population = P0 × (1 - r)^t
Substituting the values:
Population = 25000 × (1 - 0.041)^7
Calculating:
Population = 25000 × (0.959)^7
Population ≈ 25000 × 0.641
Population ≈ 16025
Therefore, the population will be approximately 16,025 in 7 years.
2) How long will it take the population to reach 15000?
Given:
Initial Population (P0) = 25000
Rate of decline (r) = 4.1% = 0.041
We need to find the time it takes for the population to reach 15000, so we'll use the formula and solve for t:
15000 = 25000 × (1 - 0.041)^t
Simplifying:
0.6 = 0.959^t
To solve for t, we can use logarithms. Taking the natural logarithm (ln) of both sides:
ln(0.6) = ln(0.959^t)
Using the property of logarithms (ln(a^b) = b × ln(a)):
ln(0.6) = t × ln(0.959)
Now, divide both sides by ln(0.959):
t ≈ ln(0.6) / ln(0.959)
Calculating:
t ≈ -0.5108 / -0.0416
t ≈ 12.28
Therefore, it will take approximately 12.28 years for the population to reach 15000.