Problem 21 - Calculus and its applications

Question

The population density of fireflies in a field is given by \(p(x, y)=\frac{1}{100} x^{2} y,\) where \(0 \leq x \leq 30\) and \(0 \leq y \leq 20, x\) and \(y\) are in yards, and \(p\) is the number of fireflies per square yard. a) Determine the total population of fireflies in this field. b) Determine the average number of fireflies per square yard of the field.

Step-by-Step Solution

Answer

The total population is 72,000 fireflies, and the average density is 120 fireflies per square yard.

Step 1: Understand the Problem

We want to determine the total population of fireflies and their average density per square yard in a specified field with limits for \(x\) from 0 to 30 and \(y\) from 0 to 20. The density function provided is given by \(p(x, y) = \frac{1}{100}x^2 y\).

Step 2: Set Up the Integral for Total Population

The total number of fireflies is found by integrating the density function \(p(x, y)\) over the given region \([0, 30] \times [0, 20]\). The integral is:\[\int_{0}^{30} \int_{0}^{20} \frac{1}{100} x^{2} y \, dy \, dx\]

Step 3: Integrate with Respect to \(y\)

Calculate the inner integral with respect to \(y\):\[\int_{0}^{20} \frac{1}{100} x^{2} y \, dy = \frac{1}{100} x^{2} \left[ \frac{1}{2} y^2 \right]_{0}^{20} = \frac{1}{100} x^{2} \cdot \frac{1}{2} \cdot 400 = \frac{1}{50} x^{2} \cdot 400 = 8x^2\]

Step 4: Integrate with Respect to \(x\)

Integrate the result with respect to \(x\):\[\int_{0}^{30} 8 x^{2} \, dx = 8 \int_{0}^{30} x^{2} \, dx = 8 \left[ \frac{1}{3} x^{3} \right]_{0}^{30} = 8 \cdot \frac{1}{3} \cdot 27000 = 72000\]

Step 5: Determine Total Population

The total population of fireflies in the field is thus 72000 fireflies. This is the result of the double integration we performed.

Step 6: Calculate Field Area

To find the average density, calculate the area of the field: Area = Width \(\times\) Height = 30 \(\times\) 20 = 600 square yards.

Step 7: Compute the Average Density

The average number of fireflies per square yard is the total population divided by the area:\[\text{Average density} = \frac{72000}{600} = 120\]

Key Concepts

Population Density

Population density refers to the number of individuals residing within a specific unit of area. In mathematical terms, it helps us understand how something is spread out over a defined space. In the context of our exercise, we are dealing with the population density of fireflies across a field. The density function provided, \( p(x, y) = \frac{1}{100}x^2y \), indicates how densely packed the fireflies are, depending on their location within the field.- The function \( p(x, y) = \frac{1}{100}x^2y \) tells us that the density increases as we move in the positive direction of the \(x\) and \(y\) axes within the limits.This makes it clear that different parts of the field can have varying densities of fireflies rather than the same everywhere. Knowing this is crucial when you are trying to calculate either the total population or other related parameters of fireflies in a given area.

Firefly Population

To find out the total firefly population in the field, we need to integrate the population density function over the entire field's area. Double integration is used here because it allows us to sum up the small patches of the field, each with its own firefly count as determined by the density function.- The integration follows this format: \[ \int_{0}^{30} \int_{0}^{20} \frac{1}{100} x^{2} y \, dy \, dx \]- This two-step process involves first integrating with respect to \(y\), followed by \(x\). The resulting value, in this case, 72000, tells us the total number of fireflies across the field.What makes calculating the firefly population interesting is that it provides insight into not only how many fireflies inhabit the field but also how their concentration might vary across the field. This approach ensures that areas with higher concentration are given appropriate weight in the calculation.

Average Density

Average density takes the total number of fireflies and divides it by the total area to find the average number of fireflies per unit area, in this case, per square yard. This calculation helps understand how thickly packed an item or living being is over an entire space.- For the field in question, the total area is found by multiplying its dimensions: \[ \text{Area} = 30 \times 20 = 600 \text{ square yards} \]- The calculated average density is therefore: \[ \text{Average density} = \frac{72000}{600} = 120 \]This means, on average, there are 120 fireflies per square yard in the field. Average density gives a generalized idea of spread when an unused observer can't delve into every detailed measure across the entire field. It's a handy measure of estimating overall density without needing a microscopic look.