To obtain the dimensions the gardener should use for the vegetable garden, we proceed as follows:
Step 1: We make a sketch of the rectangular vegetable garden
Step 2: Label the diagram with the dimensions
Since we want the rectangular vegetable garden to be 12 feet longer than it is wide, we have that:
Length = 12 feet + Width
If we represent the dimension of the Width as w, then we have:
Length = 12 feet + w
OR
Length = 12 + w
Now, the diagram becomes:
Step 3: Now, we will write the expression for the area of a rectangle
This is:
[tex]\text{Area of a rectangle = Length }\times Width[/tex]Step 4: Since we know the area of the rectangular vegetable garden to be 64 square feet, and we know the length to be (12 + w), and the width to be w, we can substitute these value and expressions in the formula for the area of a rectangle, as follows:
[tex]\begin{gathered} \text{Area of a rectangle = Length }\times Width \\ 64\text{ = }(12+w)\times w \\ \end{gathered}[/tex]Step 5: Now, we simply the expression and solve the resulting equation
[tex]\begin{gathered} 64\text{ = }(12+w)\times w \\ 64\text{ = (}12\times w)+(w\text{ }\times w) \\ 64=12w+w^2 \\ w^2+12w=64 \\ w^2+12w-64=0 \end{gathered}[/tex]Step 6: We can solve the resulting quadratic equation using any method, but here, we will be using the factorization method, as follows:
[tex]\begin{gathered} w^2+12w-64=0 \\ \text{The factors are +16 and -4} \\ \text{Thus:} \\ w^2+16w-4w-64=0 \\ w(w+16)-4(w+16)=0 \\ (w+16)(w-4)=0 \\ \text{Thus:} \\ (w+16)=0\text{ or }(w-4)=0 \\ w=-16\text{ or }w=4 \end{gathered}[/tex]Taking only the positive value of w, we have that w = 4
Step 7: We can now obtain the length, as follows:
Length = 12 + w
Length = 12 + 4
Length = 16
Therefore, the dimensions that the gardener should use for the vegetable garden are:
Length = 16
Width = 4
