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.?

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

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

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: \(\frac{18,143.7 \text{ g}}{7,192.28 \text{ cm}^3} = 2.52361 \frac{\text{g}}{\text{cm}^3}\)

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.

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.