Modelling

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Advanced Modelling Skills

Discussion

A truck accident dumped a toxic chemical into a river that feeds into the pond in Giuseppe's town. He knows that the concenctration of the chemical C can be modelled by

C(t)=16e−0.0144tmg/L,

where t is the time since the accident in hours. Giuseppe uses the model to prepare a crisis plan for the pond.

Once the concentration falls below 8mg/L, it is safe for divers to enter the pond to save the remaining fish.

When the concentration decreases to 4mg/L, the park bordering the pond may reopen.

Finally, the river also feeds into the water plant that provides the town's tap water, and the tap water will only be safe for drinking and cooking once the concentration falls to 2mg/L.

Find how long until
1. the divers can save the fish.
2. the park reopens.
3. the tap becomes safe to consume.
Do you notice anything about how long it takes for the concentration of the chemical to halve?

Part (a) (i)

We want to find the time t when the concentration drops to 8mg/L, so we solve

16e−0.0144t=8

You can use the graph-intersection method on a TI-84:

Enter Y1=16e−0.0144X and Y2=8 in the Y= menu.
Press GRAPH to display both curves.
Use 2nd TRACE (CALC) → 5:intersect, and press ENTER three times to find the intersection.

Powered by Desmos
The calculator will display X≈48.1, confirming the result.

Part (a) (ii)

We want the time t when the concentration drops to 4mg/L, so we solve

16e−0.0144t=4

On a TI-84:

Enter Y1=16e−0.0144X and Y2=4 in the Y= menu.
Press GRAPH.
Press 2nd TRACE (CALC) → 5:intersect and hit ENTER three times.

The calculator returns

X≈96.3

so it takes about 96.3 hours for the concentration to fall to 4mg/L.

Part (a) (iii)

We want to find the time t when the concentration drops to 2mg/L, so we solve

16e−0.0144t=2

You can use the graph-intersection method on a TI-84:

• Enter Y1=16e−0.0144X and Y2=2 in the Y= menu.

• Press GRAPH to display both curves.

• Use 2nd TRACE (CALC) → 5:intersect, and press ENTER three times to find the intersection.

The calculator will display

X≈144.4

so it takes about 144.4 hours for the concentration to fall to 2mg/L.

Part (b)

From part (a) we have:

16→8mg/L in 48.1h
8→4mg/L in 96.3−48.1=48.2h
4→2mg/L in 144.4−96.3=48.1h

Each time the concentration halves, it takes about 48.1h.

Thus, the chemical’s concentration halves in a constant time of approximately 48.1h.

Calculating half-life

From any expoential decay model of the form f(t)=Abkt (0<b<1), the half-life, or time for the value of f to reach half of its current value, is given by t1/2=−klogb2.

Most commonly, given an equation of the form f(t)=Aekt, the half life is given by −kln2.

Checkpoint

Find the half life of f(t)=4⋅e−3t.

Select the correct option

Discussion

Giuseppe wants to treat the river water with another chemical that reacts with the toxic chemical, reducing its concentration. He finds the following plot that graphs the recommended concentration dose D against the concentration of the toxic chemical C:

Describe how a model D(C) for the given data shoud look.

The graph of D(C) should

• Pass smoothly through all the plotted points.

• Have a vertical asymptote at C=0 (so as C→0, D(C)→−∞ or very large in magnitude).

• Increase very steeply for small positive C, matching the rapid rise of the points just to the right of the vertical axis.

• Then “level off” as C grows, with the slope decreasing so that for larger C the curve flattens and climbs only gradually.

In other words, sketch a single continuous, strictly increasing curve that shoots up sharply near C=0, and thereafter approaches a more or less horizontal trend through the higher-C points.

Natural logarithmic models

A natural logarithmic model is given by f(x)=a+blnx.

Notice f(1)=a and f(e)=a+b.

Checkpoint

Fit the logarithm to the plotted points.

Recall that we defined a sinusoidal model as f(x)=asin(bx)+c (or f(x)=acos(bx)+c). The parameters a, b, and c account for vertical stretches and reflections, horizontal stretches and reflections, and vertical translations respectively. To account for a horizontal shift of sinusoidal models, we must introduce a new parameter h.

Phase shifts

If a sinusoidal model has a phase shift, it has been moved horizontally. Now, f(x)=asin(b(x−h))+c, where h is the phase shift.

Checkpoint

Using the graph of sin(x−h) below, find h so that sin(x−h)=cos(x).

Select the correct option

Discussion

Bob plots a population count p of bacteria in a petri dish against time t (hours):

Describe a curve that fits Bob's data.

A single smooth S-shaped curve fits the scatter: it lies nearly flat at p≈0 for small t, then rises steeply, and finally levels off approaching a horizontal asymptote just under p≈550.

Logistic Models

A logistic model describes growth that appears exponential for smaller values but slows as it approaches a carrying capacity, represented by a horizontal asymptote above the curve.

Logistic models are given by the general equation f(x)=1+Ce−kxL, where L is the carrying capacity. Logistic models are particularly effective for modelling population growth, as they tend to grow exponentially from small numbers yet have a carrying capacity capped by the scarcity of space, food, and water. k is often called the intrinsic rate, and it represents the rate of growth of a quantity before it nears carrying capacity. Finally, C controls the initial population since f(0)=1+CL, where f(0) is the initial population.

Checkpoint

Fit the logistic model to the data.

Discussion

John throws a tennis ball up into the air, watches it fall, and lets it bounce until it finally comes to rest on the ground. The bouncing ball and its height versus time are shown below.

Combine modelling skills that you have already learned to describe the height of the ball as a function of time.

We observe that the complete height–time graph consists of four “flight” arcs, each one a downward‐opening parabola whose peak is lower than the one before.

Putting it all together, h(t) is a piecewise function made up of these four downward‐opening parabolas, each defined on one of the successive time‐intervals and each with a smaller top height than the previous one.

Thus, the motion of the ball combines what we learned about quadratic models and linear piecewise models!

Non-Linear Piecewise Models

On the HL test, you may see piecewise models that have non-linear pieces.

For example, f(x)={xx<0x2x≥0is graphed below.

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