Question

A farmer builds a water trough to fit a corner. The water trough is made of tworectangular prisms.What is the volume of the water trough?

Answer 1

Given the figure labeled below:

A) Length A and Width B

In the above figure,

`\begin{gathered} BC=VW+XY\text{ } \\ similarly, \\ AB=WX+YZ \end{gathered}`
`\begin{gathered} where \\ BC=8 \\ VW=3 \\ XY=A \\ thus,8=3+A \\ subtract\text{ 3 from both sides} \\ 8-3=3-3+A \\ \Rightarrow A=5\text{ ft} \end{gathered}`
`\begin{gathered} Also, \\ AB=8 \\ WX=4 \\ YZ=B \\ thus, \\ 8=4+B \\ subtract\text{ 4 from both sides of the equation} \\ 8-4=4-4+B \\ \Rightarrow B=4\text{ ft.} \end{gathered}`

Thus,

`\begin{gathered} length\text{ A = 5 ft.} \\ width\text{ B= 4 ft.} \end{gathered}`

B) The volume of the water trough:

Since the water trough is made up of two rectangular prisms, we can split the figure into two rectangular prisms as shown below:

To calculate the volume,

step 1: Evaluate the volume of prism 1.

The volume of prism 1 is expressed as

`\begin{gathered} Volume\text{ = base area}* height \\ where \\ base\text{ area = length}* width=4\text{ ft}*3\text{ ft=12 ft}^2 \\ height=2\text{ ft} \\ thus, \\ Volume\text{ of prism 1 = 12 ft}^2*2\text{ ft} \\ \Rightarrow Volume\text{ of prism 1 = 24 ft}^3 \end{gathered}`

step 2: Evaluate the volume of prism 2.

Similarly, the volume of prism 2 is evaluated as

`\begin{gathered} Volume\text{ of prism 2= \lparen length}* width)* height \\ where \\ length=8\text{ ft} \\ width=4\text{ ft.} \\ height=2\text{ ft.} \\ thus, \\ volume\text{ of prism 2 =\lparen8 ft}*4ft)*2\text{ ft} \\ \Rightarrow volume\text{ of prism 2 = 64 ft}^3 \end{gathered}`

step 3: Sum up the volumes of prisms 1 and 2.

Thus, the volume of the water trough is evaluated to be

`\begin{gathered} Volume\text{ of water trough = 24 ft}^3+\text{ 64 }ft^3 \\ \Rightarrow Volume\text{ of water trough = 88 }ft^3 \end{gathered}`