Problem 92 - General Chemistry

Question

The population density of worms in a particular field is 33 worms per cubic meter of soil. How many worms would there be in the top meter of soil in a field that has dimensions of \(1.00 \mathrm{~km}\) by \(2.0 \mathrm{~km}\) ?

Step-by-Step Solution

Answer

There are 66,000,000 worms in the top meter of soil.

Step 1: Understand the problem

We need to calculate the total number of worms in the top meter of soil in a field, given the population density of worms and the dimensions of the field.

Step 2: Convert the dimensions from kilometers to meters

Convert the dimensions of the field from kilometers to meters to work with the worm density units. Since 1 km = 1000 meters, the field measures 1000m by 2000m.

Step 3: Calculate the volume of the soil

The top meter of soil has a height of 1 meter. Calculate the volume of the soil in cubic meters using the field's dimensions: Volume = Length × Width × Height, which is 1000m × 2000m × 1m.

Step 4: Compute the volume

Multiply the dimensions to find the volume: 1000 × 2000 × 1 = 2,000,000 cubic meters.

Step 5: Calculate the number of worms

Multiply the volume of the soil by the population density to find the total number of worms: 2,000,000 cubic meters × 33 worms/m³ = 66,000,000 worms.

Key Concepts

Volume Calculation

To figure out how many worms are in the field, we first need to calculate the volume of the top meter of soil. The formula for calculating volume is:
\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \]
In this scenario, the length and width of the field are given in kilometers (km) and we want the volume in cubic meters (m³). The height of the soil we are considering is just the top meter, so that's 1 meter.

Using the formula:

• Length = 1000m (converted from 1 km)
• Width = 2000m (converted from 2 km)
• Height = 1m

Multiply these dimensions:
\[ \text{Volume} = 1000 \times 2000 \times 1 = 2,000,000 \text{ m}^3 \]
Thus, the volume of the top meter of soil in the field is 2,000,000 cubic meters.

Unit Conversion

Before doing any calculations, it is important to ensure all measurements are in the correct units. Conversion is a key step in calculations to maintain consistency and accuracy. Here we need to convert the field dimensions from kilometers to meters. By understanding that 1 kilometer equals 1000 meters, we can easily convert:

• For the length: 1 km × 1000 = 1000 meters
• For the width: 2 km × 1000 = 2000 meters

This straightforward multiplication process helps align the dimensions with the given population density, stated in worms per cubic meter. Getting the units right is crucial; it aligns all parts of the calculation ensuring accurate results.

Multiplication in Mathematics

Understanding multiplication is fundamental to solving problems like this one. Multiplication allows us to quickly calculate the total number of worms over a large area by using the given density. Here, the multiplication combines the volume of soil with the density of worms.Let's break it down:

• The field's topsoil volume is 2,000,000 m³.
• The population density is 33 worms per cubic meter (33 worms/m³).

To find the total number of worms in the top meter of soil, multiply the volume by the density:\[ 2,000,000 \text{ m}^3 \times 33 \text{ worms/m}^3 = 66,000,000 \text{ worms} \]This multiplication shows how combining density with volume provides the total count over a specified area, highlighting how fundamental arithmetic operations can solve real-world questions efficiently.