Question

g 1) The rate of growth of a certain type of plant is described by a logistic differential equation. Botanists have estimated the maximum theoretical height of such plants to be 30 in. At the beginning of an experiment, the height of a plant was 5 in. a) Find an expression for the height of the plant after t days. (To find k, suppose the plant grew to 12 in. after 20 days.) b) What was the height of the plant after 30 days

Answer 1

Answer:

a) The expression for the height, 'H', of the plant after 't' day is;

$H = \dfrac{30}{1 + 5\cdot e^{-(2.02732554 \times 10^{-3}) \cdot t}}$

b) The height of the plant after 30 days is approximately 19.426 inches

Step-by-step explanation:

The given maximum theoretical height of the plant = 30 in.

The height of the plant at the beginning of the experiment = 5 in.

a) The logistic differential equation can be written as follows;

$\dfrac{dH}{dt} = K \cdot H \cdot \left( M - {P} \right)$

Using the solution for the logistic differential equation, we get;

$H = \dfrac{M}{1 + A\cdot e^{-(M\cdot k) \cdot t}}$

Where;

A = The condition of height at the beginning of the experiment

M = The maximum height = 30 in.

Therefore, we get;

$5 = \dfrac{30}{1 + A\cdot e^{-(30\cdot k) \cdot 0}}$

$1 + A = \dfrac{30}{5} = 6$

A = 5

When t = 20, H = 12

We get;

$12 = \dfrac{30}{1 + 5\cdot e^{-(30\cdot k) \cdot 20}}$

$1 + 5\cdot e^{-(30\cdot k) \cdot 20} = \dfrac{30}{12} = 2.5$

$5\cdot e^{-(30\cdot k) \cdot 20} = 2.5 - 1 = 1.5$

∴ -(30·k)·20 = ㏑(1.5)

k = ㏑(1.5)/(30 × 20) ≈ 6·7577518 × 10⁻⁴

k ≈ 6·7577518 × 10⁻⁴

Therefore, the expression for the height, 'H', of the plant after 't' day is given as follows

$H = \dfrac{30}{1 + 5\cdot e^{-(30\times 6.7577518 \times 10^{-4}) \cdot t}} = \dfrac{30}{1 + 5\cdot e^{-(2.02732554 \times 10^{-3}) \cdot t}}$

b) The height of the plant after 30 days is given as follows

$H = \dfrac{30}{1 + 5\cdot e^{-(2.02732554 \times 10^{-3}) \cdot t}}$

At t = 30, we have;

$H = \dfrac{30}{1 + 5\cdot e^{-(2.02732554 \times 10^{-3}) \times 30}} \approx 19.4258866473$

The height of the plant after 30 days, H ≈ 19.426 in.