The Uncertainty of Volume in a Petri Dish: Calculating the Exact Measurements

In the world of scientific research, precision is paramount. This is particularly true in the field of biology, where even the smallest miscalculation can lead to significant errors in experimental results. One such area where precision is crucial is in the measurement of the volume of a Petri dish. These dishes, used to culture cells or small mosses, typically have a diameter of 100 ± 0.2 mm and a height of 15 ± 0.2 mm. But how does one calculate the exact volume of a Petri dish, taking into account the uncertainty in these measurements? Let’s delve into this intriguing topic.

Understanding the Concept of Uncertainty

Before we proceed to calculate the volume, it’s important to understand the concept of uncertainty in measurements. Uncertainty refers to the degree of doubt about a measurement, which arises due to the limitations of the measuring instrument or the method of measurement. In our case, the uncertainty in the diameter (∆d) and height (∆h) of the Petri dish is ±0.2 mm.

Calculating the Volume of a Petri Dish

The volume V of a cylindrical object like a Petri dish can be calculated using the formula V = πd²h/4, where d is the diameter and h is the height. Substituting the given values, we get V = π(100 mm)²(15 mm)/4 = 117,810 mm³ or 117.81 cm³.

Calculating the Uncertainty in Volume

Now, let’s calculate the uncertainty in volume. The fractional uncertainty in volume (∆V/V) is given by the sum of the fractional uncertainties in diameter and height, i.e., ∆V/V = ∆d/d + ∆h/h. Substituting the given values, we get ∆V/V = 0.2/100 + 0.2/15 = 0.002 + 0.0133 = 0.0153.

The absolute uncertainty in volume (∆V) can then be calculated by multiplying the fractional uncertainty by the volume, i.e., ∆V = ∆V/V * V = 0.0153 * 117,810 mm³ = 1,803 mm³ or 1.80 cm³. Therefore, the volume of the Petri dish is 117.81 ± 1.80 cm³.

Implications of Uncertainty in Volume

The uncertainty in the volume of a Petri dish may seem insignificant, but it can have profound implications in biological research. For instance, it can affect the concentration of the culture medium, which in turn can influence the growth and behavior of cells or mosses. Therefore, it’s crucial for researchers to take into account the uncertainty in volume when designing and conducting experiments.

In conclusion, the calculation of the exact volume of a Petri dish, taking into account the uncertainty in measurements, is not only a fascinating exercise in applied mathematics but also a critical aspect of precision in biological research.