Math and science offers a great tool for understanding scientific concepts. Be it academics, research, or someone’s curiosity, it has helped everyone. We discussed the linear functions in the context of mathematics for biology in the previous articles. Study of mathematical power functions aid in biological research and education.
The reach of mathematics in science beyond imagination and it is still growing. It can’t be said how farther it can go. Learning science math and the use of it in biology or any field of science would be the beginning of the bigger endeavor. In previous articles, we discussed what are the functions, linear functions, and their correspondence with biological aspects
Have you ever noticed the difference between the prices and the size of the pizza when you go to buying one? You don’t get the pizza at doubled cost if the size of the pizza is doubled. You might argue that, why you can’t get the pizza of double size in the double cost. This is obvious when one is missing out on the simple mathematics that lies behind the making of the pizza. The same goes for cakes as well.
To know the answer to the question why is that so, we need to look at the changes in the area of the pizza with changing size. The above graph depicts the relation between the surface area and the size of the pizzas available in the market. Look at the area of the 10” and 14” sized pizzas. The area of the 14” sized pizza is almost twice the area of 10” sized pizza.
Values of the area of the corresponding pizzas are marked by red color in the graph. When the diameter of the pizza is changing only by 4”, the area is doubling. It means the area of the pizza is not linearly proportional to the size of the pizza. And that is why you can not get the pizza of double size at a doubled price.
Let us look the geometry of the circle. The area of the circle is calculated as,
The area of the circle increases as the radius of the circle increases. But how much is the increase? The area of the circle will be the numerical value of 𝜋 multiplied by the square of the radius of the circle. Thus the change in the area of the circle is proportional to the square of the increased radius.
For a circular pizza, the area of the pizza almost doubles when the size of the 10’’ pizza changed to 14’’ pizza. Now it makes sense that more area of the pizza is going to need more quantity of the material for making which attracts significant addition to the cost of the pizza.
The pizzas available in the market are more or less of the same thickness, which is not the case with cakes. They come in varying widths and thicknesses. Here, thickness also adds to the material that will be needed to make the cake. The total material required for making the cake is going to be equal to the volume of the cake
For a moment let us assume that the shape of the cake is a cubicle. Thus the volume of the cubicle will be calculated as,
The volume of the cube is calculated by multiplying its sides (LXBXD). All sides are the same for a cube, thus the volume will be proportional to the length of the side raised to the power of three.
In both, the examples mentioned above the area, A, and volume V are the function of the radius of the circle and the length of the sides of the cube respectively.
This type of function is called a power function because the value of the dependent variable (Area and Volume) is dependent upon the independent variable (radius and Length) raised to a certain power. In the case of the Linear function, the power of the independent variable is one and that is why the value of the dependent variable changes linearly.
The difference between the linear function and power function:
Here X is an independent variable and Y is a dependent variable. Notice the change in the values of the dependent variable with the changing powers of the independent variable. Graphical representation of the above function makes the picture clearer.
In bio math, power functions have innumerable applications. We will discuss important technical aspects of the power functions in Power functions- Biology!-II. We will also discuss the biological significance of the power function.
