Question

A gardener wants to create a rectangular vegetable garden in a backyard. She wantsto have a total area of 64 square feet, and it should be 12 feet longer than it is wide.What dimensions should she use for the vegetable garden?

Answer 1

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:

`\text{Area of a rectangle = Length }* Width`

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:

`\begin{gathered} \text{Area of a rectangle = Length }* Width \\ 64\text{ = }(12+w)* w \\ \end{gathered}`

Step 5: Now, we simply the expression and solve the resulting equation

`\begin{gathered} 64\text{ = }(12+w)* w \\ 64\text{ = (}12* w)+(w\text{ }* w) \\ 64=12w+w^2 \\ w^2+12w=64 \\ w^2+12w-64=0 \end{gathered}`

Step 6: We can solve the resulting quadratic equation using any method, but here, we will be using the factorization method, as follows:

`\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}`

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