# How Mathematical Models are Used in Science

Instructor: Catherine Glover

Catherine has a master's degree in Mathematical Biology and teaches math at the college level.

Do you ever wonder how scientists make predictions? Instead of a crystal ball, they actually use mathematical models! In this lesson, learn about how these models are used in science.

## Mathematics to the Rescue!

Sarah is working in a laboratory testing new antibiotics. She needs to grow 10 million bacteria for her experiments. If she starts with 10,000 bacteria and grows them in a flask for 24 hours, will she have enough bacteria?

Sarah could take samples and count the bacteria using a microscope, but it's finals week, and she doesn't have time. Instead Sarah could quickly calculate how many bacteria she will have grown using a mathematical model, or mathematical representation, of bacterial growth. This will save Sarah time so that she can study for her exams. Score!

Math is a universal language, so mathematical models can be used to describe and solve problems in any scientific discipline. In this lesson, we'll explore math models used in biology, chemistry, and physics.

## Biological Growth

First, let's solve Sarah's problem. Bacteria reproduce by dividing in half. We can use this fact to model bacterial growth using an exponential growth model, which indicates that the number of bacteria grows exponentially over time. The basic form of the model is b(t) = b0*2rt.

Where:

• b(t) is the total number of bacteria at a certain time t.
• b0 is the initial condition or starting number of bacteria, which we know is 10,000.
• r is the exponential growth rate, which is a parameter, or variable specific to our situation.
• Finally, note that the base of the exponent is 2 because every bacterial division produces 2 bacteria.

Let's assume the exponential growth rate is 0.5. For Sarah's particular situation, at any time t, the model of bacterial growth is b(t) = 10,000*20.5t.

Let's see how many bacteria Sarah will have after t = 24 hours.

• b(24) = 10,000*20.5*24
• = 10,000*212
• = 10,000*4,096
• = 40,960,000.

There will be 40,960,000 bacteria after 24 hours of growth - plenty for Sarah's experiments!

## The Chemistry of Decay

Megan is a paleontologist trying to determine the age of a fossil. The fossil contains a specific type of carbon called carbon-14, and she knows that carbon-14 decays into nitrogen at an exponential rate. She found that the fossil initially contained 50 grams of carbon-14, but now it only contains 5 grams. How old is the fossil?

Since the amount of carbon-14 decays exponentially, we can use an exponential model again. But instead of using an exponential growth model, we will use an exponential decay model, which indicates that the amount of carbon-14 is decreasing exponentially over time.

The basic form of the model is similar to bacterial growth, but the exponent will be negative to indicate decay. The amount of carbon-14 in the fossil is modeled by:

c(t) = c0e-rt

We know that the fossil started with 50 grams of carbon-14 (c0 = 50). Megan read that the exponential decay rate of carbon-14 is r = 0.00012. So at any time t, the amount of carbon-14 remaining in the fossil is c(t) = 50e-0.00012t. Note that we are using the mathematical constant e = 2.718… as the base of the exponent to indicate continuous decay.

Since there's only 5 grams of carbon-14 left after t years of decay, c(t) = 5. Therefore:

• 5 = 50e-0.00012t
• 0.1 = e-0.00012t (dividing both sides of the equation by 50)
• ln(0.1) = ln(e-0.00012t) (taking the natural logarithm of both sides of the equation
• ln(0.1) = -0.00012t
• t = ln(0.1)/-0.00012
• = 19,188

The fossil is 19,188 years old!

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