The growth of stray cat population, n(t) in a small town can be modeled as n(t) = no e 0.15 t where t is time in years.
Given that in year 1995, the town had 20 stray cats.
a) Find the projected population after 5 years
b) Find the number of years required for the stray-cat population to reach 200.
(a) plug in t=5
(b) solve for t in 20e^.015t = 200
To find the projected population after 5 years, we need to substitute t = 5 into the equation n(t) = no e 0.15 t.
a) Find the projected population after 5 years:
Step 1: Substitute t = 5 into the equation.
n(5) = no e^(0.15 * 5)
= 20 * e^(0.75)
≈ 20 * 2.117000016612675
≈ 42.34
Therefore, the projected population after 5 years is approximately 42.34 stray cats.
To find the number of years required for the stray-cat population to reach 200, we need to solve the equation n(t) = 200 for t.
b) Find the number of years required for the stray-cat population to reach 200:
Step 1: Set n(t) = 200.
200 = no e^(0.15t)
Step 2: Divide both sides by no.
200/no = e^(0.15t)
Step 3: Take the natural logarithm (ln) of both sides.
ln(200/no) = ln(e^(0.15t))
Step 4: Simplify the right side using the property ln(e^x) = x.
ln(200/no) = 0.15t
Step 5: Divide both sides by 0.15.
t = (1/0.15) * ln(200/no)
t = (1/0.15) * ln(200/20)
t = (1/0.15) * ln(10)
t ≈ (6.666666666666667) * 2.302585092994046
t ≈ 15.384615384615383
Therefore, it would take approximately 15.38 years for the stray-cat population to reach 200.