Final answer:
To minimize the combined area of the garden and borders, we need to find the dimensions of the garden that maximize the garden's area. By expressing the area of the garden with the grass borders and setting it equal to the area of the garden, we can find the values of L and W that minimize the combined area. The dimensions of the garden will be in meters because the area is given in square meters.
Step-by-step explanation:
To minimize the combined area of the garden and borders, we need to find the dimensions of the garden that maximize the garden's area. Let's assume the length of the garden is L and the width of the garden is W.
The area of the garden is given by A = LW.
To find the dimensions that minimize the combined area, we need to express the area of the garden including the grass borders.
The length of the garden with the grass borders is L + 1 + 1 = L + 2. Similarly, the width of the garden with the grass borders is W + 2 + 2 = W + 4.
The area of the garden with the grass borders is (L + 2)(W + 4). We want to minimize this area while keeping the area of the garden fixed, which is LW = 646m².
Therefore, the combined area is given by (L + 2)(W + 4) = LW + 4L + 2W + 8.
Substituting LW = 646m², we have 646 + 4L + 2W + 8.
By differentiating the equation with respect to L and setting it equal to zero, we can find the value of L that minimizes the combined area. Similarly, we can differentiate with respect to W to find the value of W that minimizes the combined area.
Once we have the values of L and W, we can calculate the combined area.
It is important to note that since the area is given in square meters, the dimensions of the garden will also be in meters.