Question

Pheromones are compounds secreted by females of many insect species to attract mates. Typically, \(1.0 \times 10^{-8} \mathrm{~g}\) of a pheromone is sufficient to reach all targeted males within a radius of \(0.50 \mathrm{mi}\). Calculate the density of the pheromone (in grams per liter) in a cylindrical air space having a radius of \(0.50 \mathrm{mi}\) and a height of \(40 \mathrm{ft}\) (volume of a cylinder of radius \(r\) and height \(h\) is \(\pi r^{2} h\) ).

Short answer

The density is approximately \(4.02 \times 10^{-19} \text{ g/L}\).

Step-by-step solution

Convert Measurements to Consistent Units

First, let's convert the radius and height to a consistent unit. We will use meters for radius and height, then calculate the volume in liters.1 mile is approximately 1609.34 meters, so the radius is:\[0.50 \text{ mi} \times 1609.34 \text{ m/mi} = 804.67 \text{ m}\]1 foot is approximately 0.3048 meters, so the height h is:\[40 \text{ ft} \times 0.3048 \text{ m/ft} = 12.192 \text{ m} \]

Calculate the Volume of the Cylinder

Using the formula for the volume of a cylinder, which is \(V = \pi r^2 h\), calculate the volume. - Radius \( r = 804.67 \text{ m} \) - Height \( h = 12.192 \text{ m} \)\[ V = \pi \times (804.67)^2 \times 12.192 \approx 24899201.73 \text{ m}^3 \]

Convert Volume from Cubic Meters to Liters

Since 1 cubic meter is equal to 1000 liters, convert the volume calculated in cubic meters to liters.\[ V = 24899201.73 \text{ m}^3 \times 1000 \text{ L/m}^3 = 24899201730 \text{ L}\]

Calculate the Density of the Pheromone

Density is calculated as mass divided by volume. Given the mass of the pheromone is \(1.0 \times 10^{-8} \text{ g}\), use the volume in liters calculated previously.\[ \text{Density} = \frac{1.0 \times 10^{-8} \text{ g}}{24899201730 \text{ L}} \approx 4.02 \times 10^{-19} \text{ g/L} \]

Conclusion

The density of the pheromone in the defined air space is approximately \(4.02 \times 10^{-19} \text{ g/L}\).

Key concepts

cylinder volume

A cylinder is a three-dimensional shape with two parallel circles connected by a curved surface. To find the volume of a cylinder, use the formula \(V = \pi r^2 h\). Here, \(r\) is the radius of the base of the cylinder, and \(h\) is the height. This formula essentially calculates how much space is inside the cylinder. The term \(\pi r^2\) computes the area of the circular base, and multiplying by \(h\) extends this area through the height of the cylinder. In practical terms, this could help determine how much liquid or air might fit inside a cylindrical container. In the context of pheromones spread in a cylindrical region of air, calculating the volume helps in understanding the space in which the pheromone is dispensed. This allows us to further compute how concentrated the pheromone is within that volume.

unit conversion

Unit conversion is essential when different measurements are given in varying units. To effectively calculate quantities like volume, density, or other properties, we need consistent units. Here, we are converting distances from miles and feet to meters to maintain uniformity.

• 1 mile is approximately 1609.34 meters, so a radius of 0.50 miles is converted to 804.67 meters.
• 1 foot is equivalent to 0.3048 meters, turning a height of 40 feet into 12.192 meters.

These conversions simplify the process of using mathematical equations, as they ensure that all dimensions are described using the same units. Once converted to a common unit, calculations concerning volume or other measurements can proceed without error.

density calculation

Density is a measure of how much mass is contained in a certain volume. It is calculated by dividing an object's mass by its volume, given by the formula:\(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\).In this exercise, the pheromone’s density is calculated in terms of grams per liter. With the pheromone mass given as \(1.0 \times 10^{-8} \text{ g}\) and the volume found to be 24899201730 liters, the density computation follows:\(\text{Density} = \frac{1.0 \times 10^{-8} \text{ g}}{24899201730 \text{ L}} \approx 4.02 \times 10^{-19} \text{ g/L}\).Understanding what this tiny density number signifies helps comprehend how sparse and distributed the pheromones are in the air space. Although the mass quantity is minuscule, it is sufficient to affect the targeted receivers due to the specific nature of pheromone function in insect communication.

pheromones in insect communication

Pheromones play a critical role in the communication systems of many insects. They are chemicals secreted by individuals which influence the behavior or development of other members of the same species. In sex pheromones, like in our example, females secrete these compounds to signal their presence to males, often over long distances. This form of communication is highly effective despite the pheromone being in very low concentration in the air.

• Pheromones are detected by receptor organs specialized for chemical detection, located typically on the antennae of insects.
• Even the slightest amount of pheromone can trigger behavioral responses, ensuring mating success even in large and open environments.

Insect pheromones, thus, exemplify how chemical communication can efficiently operate within highly dispersed environments, confirming the importance of understanding how such compounds distribute themselves in air spaces.