Question

1. a garden bed, pictured below, is a square prism with a rectangular prism taken out. the inside length of the square prism is 6 feet. the rectangular prism taken out has a width of 2 feet and a length of 4 feet. the diagram below shows the top - view of the garden bed with its inside measurements.

the garden bed is filled with topsoil to a uniform height of 1.25 feet.
determine and state the volume of the topsoil, in cubic feet.
each bag of topsoil sells for $3.58 and contains 2 cubic feet of topsoil.
determine and state the total cost of the bags of topsoil that must be purchased to fill the garden.

Explanation:

Step1: Calculate volume of outer - square prism

The outer - square prism has side length \(s = 6\) feet and height \(h=1.25\) feet. The volume of a square prism \(V_{outer}=s^{2}\times h\). So \(V_{outer}=6^{2}\times1.25=36\times1.25 = 45\) cubic feet.

Step2: Calculate volume of inner - rectangular prism

The inner - rectangular prism has length \(l = 4\) feet, width \(w = 2\) feet and height \(h = 1.25\) feet. The volume of a rectangular prism \(V_{inner}=l\times w\times h\). So \(V_{inner}=4\times2\times1.25=10\) cubic feet.

Step3: Calculate volume of top - soil

The volume of the top - soil \(V = V_{outer}-V_{inner}\). So \(V=45 - 10=35\) cubic feet.

Step4: Calculate number of bags

Each bag contains 2 cubic feet of top - soil. The number of bags \(n=\frac{V}{2}=\frac{35}{2}=17.5\). Since we can't buy half - bags, we need to buy 18 bags.

Step5: Calculate total cost

Each bag costs \(\$3.58\). The total cost \(C=n\times3.58\). So \(C = 18\times3.58=\$64.44\).

Answer:

The volume of the top - soil is 35 cubic feet and the total cost of the bags of top - soil is \(\$64.44\).