Question

justin built a compost bin in the shape of a rectangular prism. the bin is 8 ft long, 4 ft wide, and 3 ft deep. after the compost cycle is complete, the bin will be full of potting soil that justin can sell at a farmer’s market. the potting soil will be packaged in bags. the amount of soil each bag can hold is known in cubic inches.
(a) find the volume of the compost bin in cubic inches. use the table of conversion facts, as needed.
(b) justin is going to put the potting soil in bags. each bag holds 630 in³ of the soil. he is going to bag up as much soil as possible, but he won’t partially fill any bags. how many whole bags will he fill?
(c) the bags will sell for $7.05 each. if justin sells all the bags, how much money will he collect?
conversion facts for length
1 foot (ft) = 12 inches (in)
1 yard (yd) = 3 feet (ft)
1 yard (yd) = 36 inches (in)

Explanation:

Part (a)

Step1: Convert length, width, depth to inches

Length: \( 8\ \text{ft} \times 12\ \text{in/ft} = 96\ \text{in} \)
Width: \( 4\ \text{ft} \times 12\ \text{in/ft} = 48\ \text{in} \)
Depth: \( 3\ \text{ft} \times 12\ \text{in/ft} = 36\ \text{in} \)

Step2: Calculate volume of rectangular prism

Volume \( V = \text{length} \times \text{width} \times \text{depth} \)
\( V = 96 \times 48 \times 36 \)
\( V = 96 \times 1728 \)
\( V = 165888\ \text{cubic inches} \)

Step1: Divide total volume by bag volume

Total volume from (a): \( 165888\ \text{in}^3 \)
Bag volume: \( 630\ \text{in}^3 \)
Number of bags \( = \frac{165888}{630} \approx 263.31 \)

Step2: Take whole number of bags

Since we can’t partially fill bags, we take the floor of \( 263.31 \), which is \( 263 \).

Step1: Multiply number of bags by price per bag

Number of bags: \( 263 \)
Price per bag: \( \$7.05 \)
Total money \( = 263 \times 7.05 \)
\( 263 \times 7 = 1841 \), \( 263 \times 0.05 = 13.15 \)
Total \( = 1841 + 13.15 = 1854.15 \)

Answer:

\( 165888 \)

Part (b)