Question

A 40 -lb container of peat moss measures \(14 \times 20 \times 30\) in. A 40 -lb container of topsoil has a volume of 1.9 gal. (a) Calculate the average densities of peat moss and topsoil in units of\(\mathrm{g} / \mathrm{cm}^{3} .\) Would it be correct to say that peat moss is "lighter" than topsoil? (b) How many bags of peat moss are needed to cover an area measuring 15.0 \(\mathrm{ft} \times 20.0 \mathrm{ft}\) to a depth of 3.0 in.?

Short answer

The average densities of peat moss and topsoil are 0.131791 g/cm³ and 2.52361 g/cm³, respectively. Since the density of peat moss is lower than that of topsoil, it is correct to say that peat moss is "lighter" than topsoil. To cover an area measuring 15.0 ft x 20.0 ft to a depth of 3.0 inches, 16 bags of peat moss are needed.

Step-by-step solution

Calculate the volume of the peat moss container in cubic inches

To do this, multiply the given dimensions of the container: \(14 \times 20 \times 30 = 8,400\) cubic inches.

Convert the volume of the peat moss container to gallons

There are approximately 231 cubic inches in a gallon. Divide the volume in cubic inches by this conversion factor: \( \frac{8400 \text{ in}^3}{1 \text{ gal}} \times \frac{1 \text{ gal}}{231 \text{ in}^3} = 36.3636\ text{ gal} \)

Calculate the average densities of peat moss and topsoil in g/cm³

First, we need to convert the weights of the peat moss and topsoil containers from pounds to grams: \(40 \text{ lb} \times \frac{453.592 \text{ g}}{1 \text{ lb}} = 18,143.7 \text{ g} \) Next, we need to convert the volumes of the peat moss and topsoil containers from gallons to cubic centimetres: - Peat moss: \(36.3636 \text{ gal} \times \frac{3,785.41 \text{ cm}^3}{1 \text{ gal}} = 137,590.079 \text{ cm}^3\) - Topsoil: \(1.9 \text{ gal} \times \frac{3,785.41 \text{ cm}^3}{1 \text{ gal}} = 7,192.28 \text{ cm}^3\) Now, we can calculate the average densities: - Peat moss: \(\frac{18,143.7 \text{ g}}{137,590.079 \text{ cm}^3} = 0.131791 \frac{\text{g}}{\text{cm}^3}\) - Topsoil: \(\frac{18,143.7 \text{ g}}{7,192.28 \text{ cm}^3} = 2.52361 \frac{\text{g}}{\text{cm}^3}\)

Determine if peat moss is "lighter" than topsoil

Since the average density of peat moss (0.131791 g/cm³) is lower than the average density of topsoil (2.52361 g/cm³), it is correct to say that peat moss is "lighter" than topsoil.

Calculate how many bags of peat moss are needed to cover the area

First, convert the given dimensions of the area from feet to inches: - Length: \(15.0 \text{ ft} \times \frac{12 \text{ in}}{\text{ft}} = 180 \text{ in}\) - Width: \(20.0 \text{ ft} \times \frac{12 \text{ in}}{\text{ft}} = 240 \text{ in}\) Calculate the total volume to be covered: \(180 \text{ in} \times 240 \text{ in} \times 3.0 \text{ in} = 129,600 \text{ in}^3\) Convert the volume to gallons: \(\frac{129,600 \text{ in}^3}{231 \text{ in}^3/\text{gal}} = 561.039 \text{ gal}\) Finally, divide the required volume by the volume of one bag of peat moss: \(\frac{561.039 \text{ gal}}{36.3636 \text{ gal/bag}} = 15.4237 \text{ bags}\) Since we can't have a fraction of a bag, we need to round up to the nearest whole number: 16 bags of peat moss are needed to cover the area.

Key concepts

Peat moss density

Density is a measure of how much mass is contained in a given volume. Peat moss, often used in gardening and horticulture, has a notably low density compared to other materials like topsoil. This is because peat moss is composed mainly of partially decayed organic matter that is less compact. When we calculated the density of peat moss, we used its weight in grams and its volume in cubic centimeters. A 40-lb container of peat moss weighs approximately 18,143.7 grams and occupies about 137,590.079 cubic centimeters in volume. After dividing the mass by the volume, the resulting density is 0.131791 g/cm³. This low density indicates that peat moss is very lightweight, making it an excellent choice for improving soil aeration and moisture retention.

Topsoil density

Topsoil, the uppermost layer of soil, is typically denser than peat moss because it contains a mix of organic material and mineral particles like sand, silt, and clay. To find the density of topsoil, we consider a 40-lb container, which is equivalent to 18,143.7 grams in weight. Because its volume is smaller at 7,192.28 cubic centimeters (coming from a volume of 1.9 gallons), the density of topsoil is calculated by dividing its mass by the volume, resulting in a density of 2.52361 g/cm³. This higher density means topsoil is heavier and more solid compared to peat moss, which is why it is often described as 'heavy soil' and is better suited for supporting plant stability and nutrient retention.

Unit conversion

Unit conversion is an essential step in many calculations, especially when dealing with different measurement systems. One common conversion is between pounds and grams. For instance, when working with weights, there are approximately 453.592 grams in a pound. Therefore, a 40-lb object can be converted to grams by multiplying by this conversion factor.
Another important conversion involves volume. Since volumes can be given in cubic inches and gallons, it is useful to know that there are 231 cubic inches in a gallon. Moreover, gallons can be further converted to cubic centimeters, with each gallon roughly equating to 3,785.41 cubic centimeters. Performing these conversions ensures consistency and accuracy when comparing and calculating measurements in scientific and everyday contexts.

Volume calculation

Volume calculation is key to understanding how much space an object occupies or how much material is needed to cover a particular area. To calculate volume, especially for rectangular shapes like containers, you multiply the object's length by its width and height. For example, the peat moss container's volume is found using its dimensions: 14 in, 20 in, and 30 in, resulting in a volume of 8,400 cubic inches.
When needing to apply a material over an area, the required volume can be determined by multiplying the area’s length, width, and depth. For instance, when you need to cover an area of 15 ft by 20 ft to a depth of 3 inches, convert these dimensions into consistent units (inches) and find the total volume. This calculation allows you to determine how many units or containers of a product you will need, which is particularly useful in landscaping and construction projects.