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# A population of aphids grows according to the law of natural growth. ?

A population of aphids grows according to the law of natural growth. If the population

size is 305 after t = 10 hours and 1847 after t = 100 hours. Determine the growth

contast k for this population. Round your final answer correctly to two decimal places.

Update:

The following scenario is used to answer the last four questions.

Use Desmos to model a population that follows a logistic growth model: f(x)=500/4e^(-0.01x) +1

Update 2:

1. What is the equation of the horizontal asymptote?

2. What does the horizontal asymptote represent?(a) the carrying capacity(b) the final population size(c) the number of time steps(d) the initial population size

### 1 Answer

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law of population growth: dy/dt = ky → k > 0

k*dt = dy/y → integrate both sides

kt + C = lny → where C is the constant of integration

y = e^[(kt+C)] = e^(kt) * e^(C) = y₀ * e^(kt)

The boundary conditions yield two equations:

305 = y₀ * e^(10k)

1847 = y₀ * e^(100k)

If we divide the first by the second, we get

305/1847 = e^(10k) / e^(100k)

0.16513 = e^(-90k)

ln(0.16513) = -1.801 = -90k

k = 0.0200112 ≈ 0.02 ◄

Bonus:

Now use either condition to find y₀:

305 = y₀ * e^(0.02*10) = y₀ * 1.22

y₀ = 250 ◄

Check using the other condition:

250 * e^(0.02*100) = 1845 → close enough!

I cannot proceed with the Desmos part. For one thing, your function is highly ambiguous. Without any parens it could mean

f(x)=(500/4)*e^(-0.01x) +1, or

f(x)=500/[4e^(-0.01x)] +1, or

f(x)=500/[4e^(-0.01x) +1]

Second, most of us (including myself) don't have an account there.

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