A \(40-1 b\) 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.?

First, we need to convert all given measurements into appropriate metric units. We will convert inches to centimeters, pounds to grams, and gallons to cubic centimeters using the conversion factors: 1 inch = 2.54 cm, 1 lb = 453.6 g, and 1 gal = 3785.41 cm³. For the peat moss container (dimensions in inches): \(14 \times 20 \times 30\) in. In centimeters: \(14 \times 2.54 \times 20 \times 2.54 \times 30 \times 2.54\) cm³ For the peat moss mass (in pounds): 40.1 lb In grams: \(40.1 \times 453.6\) g For the topsoil container (volume in gallons): 1.9 gal In cubic centimeters: \(1.9 \times 3785.41\) cm³ For the topsoil mass (in pounds): 40 lb In grams: \(40 \times 453.6\) g

Density is defined as mass per unit volume. Thus, we can calculate the density of both peat moss and topsoil using their mass and volume (in grams and cm³, respectively). Density of peat moss = \(\frac{\text{Mass of peat moss}}{\text{Volume of peat moss container}} = \frac{40.1 \times 453.6}{14 \times 2.54 \times 20 \times 2.54 \times 30 \times 2.54}\) g/cm³. Density of topsoil = \(\frac{\text{Mass of topsoil}}{\text{Volume of topsoil container}} = \frac{40 \times 453.6}{1.9 \times 3785.41}\) g/cm³.

Now that we have calculated the average densities of peat moss and topsoil, we can compare them to determine if peat moss is lighter than topsoil: If the average density of peat moss is less than that of topsoil, then it is correct to say that peat moss is lighter than topsoil.

To calculate the number of bags of peat moss required to cover an area of \(15.0 \mathrm{ft} \times 20.0 \mathrm{ft}\) to a depth of 3.0 in., we need to first convert the dimensions to appropriate metric units: 1 ft = 30.48 cm and 1 in = 2.54 cm Area = \(15.0 \times 30.48 \times 20.0 \times 30.48\) cm². Depth = \(3.0 \times 2.54\) cm. Now, we will multiply the area by the depth to find the volume required and then divide this volume by the volume of one bag of peat moss to find the number of bags: Volume required = Area × Depth Number of bags = \(\frac{\text{Volume required}}{\text{Volume of one bag of peat moss}}\). Now we can calculate the number of bags required to cover the given area at the specified depth.