IDNLearn.com makes it easy to find answers and share knowledge with others. Get prompt and accurate answers to your questions from our experts who are always ready to help.
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
