Start by finding the volume of the topsoil, which is modeled by a rectangular prism.
Buying in bulk is less expensive. See solution.
Practice makes perfect
To start off, we will need to determine how much topsoil will be needed — its volume. The volume of the topsoil can be modeled by a rectangular prism. Let's recall the formula to find the volume of a rectangular prism.
V= l w h
The length and width of the topsoil will be the same as the length and width of the backyard, 70 feet and 60 feet. We are given the length of the topsoil in inches, so we will need to convert it into feet so all components are in the same units. Converting between feet (ft) and inches (in.) will involve using a conversion factor.
1 ft/12 in.
Multiplying 4 inches by this conversion factor will convert it to feet.
The height of the topsoil is 13 feet. Let's substitute our values into the formula for the volume of a rectangular prism to find the volume of the topsoil.
V= l w h
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V=( 70)( 60)( 1/3)=1400 ft^3
The volume of topsoil needed is 1400 cubic feet.
Cost of Bags
Let's first find the cost of topsoil if bags are purchased. We will first find the number of bags needed by dividing the total volume of topsoil needed by the 3 cubic feet of topsoil per bag.
Number of bags needed=Volume of topsoil needed/Topsoil per bag
Your aunt would need 467 three cubic foot bags to landscape her backyard. To find the cost of this option, we will multiply the cost per bag by the total number of bags needed.
Cost of bags option=Number of bags needed * Cost per bag
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Cost of bags option=467 * $2.50=$1167.50
The cost for bags of topsoil is $1167.50.
Cost of Bulk
Let's find the cost of buying topsoil in bulk. The cost of topsoil is given as $22.00 per cubic yard. Let's first convert cubic yards to cubic feet so that the unit is the same as the volume of the topsoil.
1 yd^3=(1 yd)^3=(3 ft)^3=3^3 ft^3= 27 ft^3
The cost of bulk topsoil is $ 22.00 per 27 cubic feet of bulk. Let's first find the number of 27 cubic feet bulk quantities needed by dividing the total volume of topsoil needed by the 27 cubic feet of topsoil per bulk.
Amount of bulk needed=Volume of topsoil needed/Topsoil per bulk
Your aunt will need 52 cubic yards of bulk. To find the cost of this option, we will multiply the cost per cubic yard by the total number of cubic yards needed. Then we will add the $ 20 delivery fee.
Cost of bulk pre-delivery=Number of yd^3 needed * Cost per yd^3
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Cost of bulk pre-delivery=52 * $ 22.00= $1144
The cost of the bulk option before delivery is $ 1144. Finally, to find the cost of the bulk option we will add the delivery fee.
Cost of bulk option=Cost of bulk pre-delivery+Delivery fee
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Cost of bulk option=$1144+$20= $1164
The total cost of buying bulk topsoil is $1164. The cost of buying bags of topsoil is $1167.50, therefore the bulk option is less expensive.
