We are asked to find the dimensions of a rectangular flower garden that minimize the combined area of the garden and the surrounding borders.

Step 1: Define Variables

Let the dimensions of the garden be:

• $x$ = shorter edge (height) of the garden.
• $y$ = longer edge (width) of the garden.

The area of the garden is given as:

$A_{\text{garden}} = x \cdot y = 494494 \, \text{m}^2.$

Step 2: Account for the Borders

The garden has borders of different widths:

• 1 meter wide along two sides (let's assume the shorter edges, so the total length increases by 2 meters: 1 meter on each side).
• 2 meters wide along the other two sides (the longer edges, increasing by 4 meters in total: 2 meters on each side).

Thus, the overall dimensions of the area including the borders are:

• Total length = $x + 2$ meters.
• Total width = $y + 4$ meters.

Step 3: Write the Total Area (Garden + Borders)

The total area, including the garden and the borders, is:

$A_{\text{total}} = (x + 2)(y + 4).$

Step 4: Express $y$ in Terms of $x$

From the garden area equation $x \cdot y = 494494$, solve for $y$:

$y = \frac{494494}{x}.$

Step 5: Substitute $y$ into the Total Area Expression

Now substitute this value of $y$ into the total area expression:

$A_{\text{total}} = (x + 2) \left( \frac{494494}{x} + 4 \right).$

Step 6: Minimize the Total Area

To minimize the total area, take the derivative of $A_{\text{total}}$ with respect to $x$, and then find the critical points by setting the derivative equal to zero.

Derivative of $A_{\text{total}}$:

First, expand the expression:

$A_{\text{total}} = (x + 2) \left( \frac{494494}{x} \right) + (x + 2)(4).$

Simplifying:

$A_{\text{total}} = 494494 + \frac{989988}{x} + 4x + 8.$

Now, take the derivative:

$\frac{dA_{\text{total}}}{dx} = -\frac{989988}{x^2} + 4.$

Set this derivative equal to zero to find the critical points:

$-\frac{989988}{x^2} + 4 = 0.$

Solve for $x$:

$\frac{989988}{x^2} = 4,$

$x^2 = \frac{989988}{4} = 247497,$

$x = \sqrt{247497} = 7\sqrt{2}.$

Thus, the shorter edge $x$ that minimizes the total area is:

$x = 7\sqrt{2}.$

This is the height (shorter edge) of the garden.

Would you like further details or have any questions?

5 Related Questions:

1. How do you find the total width $y$ using this $x$-value?
2. How do we confirm if this $x$-value is a minimum point?
3. What would be the total combined area of the garden and borders for this optimal size?
4. How does the border size affect the overall minimization problem?
5. How can this method be generalized for a different shaped garden?

Tip:

In optimization problems, always express one variable in terms of the other to simplify the total area or volume, then differentiate to find critical points.