Exponential Growth/Decay Applications & Intro To Logistic And Cooling Models

1) A quantity changes at a rate proportional to itself and is currently decreasing. What is the sign of k in dy/dt = ky?

K > 0

K = 0 ‘

K < 0

Cannot determine

If the quantity is decreasing, dy/dt < 0 when y > 0. In dy/dt = ky, this requires k to be negative so that the product ky is negative when y is positive.

Explanation

If the quantity is decreasing, dy/dt < 0 when y > 0. In dy/dt = ky, this requires k to be negative so that the product ky is negative when y is positive.

2) The general solution to dy/dt = -0.04 y is which of the following?

Y = Ce^0.04t

Y = Ce^(-0.04t)

Y = C / (1 + 0.04t)

Y = 100 e^(-0.04t)

Separate variables: dy/y = -0.04 dt. Integrate: ln|y| = -0.04t + C. Exponentiate: y = e^C e^(-0.04t). The constant e^C is arbitrary and written as C (can be positive or negative depending on initial condition), giving y = C e^(-0.04t).

Explanation

Separate variables: dy/y = -0.04 dt. Integrate: ln|y| = -0.04t + C. Exponentiate: y = e^C e^(-0.04t). The constant e^C is arbitrary and written as C (can be positive or negative depending on initial condition), giving y = C e^(-0.04t).

3) Caffeine in the body is eliminated at a rate proportional to the amount present. If the half-life is 6 hours, what fraction remains after 18 hours?

1/2

1/4

1/8

1/6

Half-life of 6 hours means every 6 hours the amount is halved. After 18 hours = three half-lives, so the amount is multiplied by (½)³ = 1/8 of the original amount.

Explanation

Half-life of 6 hours means every 6 hours the amount is halved. After 18 hours = three half-lives, so the amount is multiplied by (½)³ = 1/8 of the original amount.

4) A town’s population grows continuously at 1.5% per year. How long does it take to double?

About 46.2 years

About 50 years

About 66.7 years

100 years

Doubling means y = 2 y0, so 2 y0 = y0 e^(0.015 t) → 2 = e^(0.015 t) → ln 2 = 0.015 t → t = ln 2 / 0.015 ≈ 0.693147 / 0.015 ≈ 46.21 years.

Explanation

Doubling means y = 2 y0, so 2 y0 = y0 e^(0.015 t) → 2 = e^(0.015 t) → ln 2 = 0.015 t → t = ln 2 / 0.015 ≈ 0.693147 / 0.015 ≈ 46.21 years.

5) The amount of medicine in a patient drops from 400 mg to 100 mg in 8 hours. When will it drop below 25 mg?

12 hours

16 hours

20 hours

24 hours

From 400 to 100 is dividing by 4 in 8 hours, so the decay factor every 8 hours is 1/4. To go from 400 to 25 is dividing by 16 = 4², so it takes two intervals of 8 hours → 16 hours total.

Explanation

From 400 to 100 is dividing by 4 in 8 hours, so the decay factor every 8 hours is 1/4. To go from 400 to 25 is dividing by 16 = 4², so it takes two intervals of 8 hours → 16 hours total.

6) In exponential growth dy/dt = ky, what happens to the solution as t → ∞ when k > 0?

Y approaches 0

Y approaches a constant

Y grows without bound

Y oscillates

The solution y = y0 e^kt with k > 0 increases exponentially as t increases, and since the exponent kt → ∞, y → ∞ with no upper limit.

Explanation

The solution y = y0 e^kt with k > 0 increases exponentially as t increases, and since the exponent kt → ∞, y → ∞ with no upper limit.

7) A bank account earns 4% interest compounded continuously. How long does it take for the balance to triple?

17.3 years

25 years

27.5 years

30 years

Tripling means final = 3 * initial, so 3 P = P e^(0.04 t) → 3 = e^(0.04 t) → ln 3 = 0.04 t → t = ln 3 / 0.04 ≈ 1.0986 / 0.04 ≈ 27.465 years ≈ 27.5 years.

Explanation

Tripling means final = 3 * initial, so 3 P = P e^(0.04 t) → 3 = e^(0.04 t) → ln 3 = 0.04 t → t = ln 3 / 0.04 ≈ 1.0986 / 0.04 ≈ 27.465 years ≈ 27.5 years.

8) Which verbal description matches dy/dt = -0.06 (80 - y)?

Exponential decay

Newton's law of cooling toward 80

Logistic growth

Pure exponential growth

The rate is proportional to the difference between y and 80 with a negative sign, meaning y approaches 80 from above (cooling) or below (heating). This is exactly Newton's law of cooling/heating.

Explanation

The rate is proportional to the difference between y and 80 with a negative sign, meaning y approaches 80 from above (cooling) or below (heating). This is exactly Newton's law of cooling/heating.

9) In the logistic equation dy/dt = 0.02 y (500 - y), what is the carrying capacity?

0.02

500

1000

Cannot determine

The standard logistic form is dy/dt = k y (L - y) where L is the carrying capacity. Comparing to 0.02 y (500 - y), the constant in front of -y is 500, so the carrying capacity L = 500.

Explanation

The standard logistic form is dy/dt = k y (L - y) where L is the carrying capacity. Comparing to 0.02 y (500 - y), the constant in front of -y is 500, so the carrying capacity L = 500.

10) A logistic population model has equation dy/dt = 0.1 y (1 - y/1000). When is the growth rate zero?

Only when y = 0

Only when y = 1000

When y = 0 or y = 1000

Never

Set dy/dt = 0: 0.1 y (1 - y/1000) = 0 → either y = 0 or 1 - y/1000 = 0 → y = 1000. These are the equilibrium points where the population stops changing.

Explanation

Set dy/dt = 0: 0.1 y (1 - y/1000) = 0 → either y = 0 or 1 - y/1000 = 0 → y = 1000. These are the equilibrium points where the population stops changing.

11) For logistic growth dy/dt = k y (L - y) with L > 0 and k > 0, which population size gives the maximum growth rate?

Y = 0

Y = L/4

Y = L/2

Y = L

The growth rate is k y (L - y) = k(Ly - y²). This is a downward-opening parabola with a vertex at y = L/2 (vertex formula for -y² + Ly is y = L/2). The maximum occurs halfway to carrying capacity.

Explanation

The growth rate is k y (L - y) = k(Ly - y²). This is a downward-opening parabola with a vertex at y = L/2 (vertex formula for -y² + Ly is y = L/2). The maximum occurs halfway to carrying capacity.

12) A species introduced to an island follows logistic growth with carrying capacity 800. At what population will it be increasing fastest?

200

400

600

800

Fastest growth in logistic model occurs at half the carrying capacity. Half of 800 is 400 individuals.

Explanation

Fastest growth in logistic model occurs at half the carrying capacity. Half of 800 is 400 individuals.

13) In a logistic model, the population starts at y(0) = 200 with carrying capacity 1000. What can we say without solving the equation?

It will exceed 1000 eventually

It will approach 1000 from below

It will decrease to zero

It will oscillate

Starting below carrying capacity (200 0 when 0 1000, the population increases toward 1000 but never crosses it.

Explanation

Starting below carrying capacity (200 0 when 0 1000, the population increases toward 1000 but never crosses it.

14) Which differential equation represents logistic growth?

Dy/dt = 0.05 y

Dy/dt = -0.05 y

Dy/dt = 0.05 y (300 - y)

Dy/dt = 0.05 (300 - y)

Logistic growth requires the rate to be proportional to both the current population y and the remaining room (C - y). Only choice C has the product y (300 - y) with positive constant.

Explanation

Logistic growth requires the rate to be proportional to both the current population y and the remaining room (C - y). Only choice C has the product y (300 - y) with positive constant.

15) A logistic model has carrying capacity 2000. The population is currently 1900 and still increasing. What will happen next?

It will keep increasing forever

It will level off near 2000

It will start decreasing immediately

It will oscillate around 2000

When y 0 so it continues increasing, but as y gets closer to 2000, (2000 - y) gets smaller, making the growth rate approach zero. The population approaches 2000 asymptotically from below.

Explanation

When y 0 so it continues increasing, but as y gets closer to 2000, (2000 - y) gets smaller, making the growth rate approach zero. The population approaches 2000 asymptotically from below.

Exponential Growth/Decay Applications & Intro to Logistic and Cooling Models