a rectangular flower garden with an area of 418 m² is surrounded by a grass border 1 m wide on two sides and…

a rectangular flower garden with an area of 418 m² is surrounded by a grass border 1 m wide on two sides and a 2 m wide on the other two sides as shown in the figure. what dimensions of the garden minimize the combined area of the garden and borders? the shorter edge (rectangular height) of the garden that will minimize the total area is √209 m. (type an exact answer, using radicals as needed.) the longer edge (rectangular width) of the garden that will minimize the total area is m. (type an exact answer, using radicals as needed.)

a rectangular flower garden with an area of 418 m² is surrounded by a grass border 1 m wide on two sides and a 2 m wide on the other two sides as shown in the figure. what dimensions of the garden minimize the combined area of the garden and borders? the shorter edge (rectangular height) of the garden that will minimize the total area is √209 m. (type an exact answer, using radicals as needed.) the longer edge (rectangular width) of the garden that will minimize the total area is m. (type an exact answer, using radicals as needed.)

Answer

Explanation:

Step1: Let the length of the garden be $x$ and the width be $y$.

We know that the area of the garden $A = xy=418$, so $y=\frac{418}{x}$.

Step2: Calculate the combined - area function.

The combined area $A_{total}=(x + 4)(y+2)$. Substitute $y=\frac{418}{x}$ into it: $A_{total}=(x + 4)(\frac{418}{x}+2)=418+2x+\frac{1672}{x}+8=426+2x+\frac{1672}{x}$.

Step3: Find the derivative of the combined - area function.

Differentiate $A_{total}(x)$ with respect to $x$. $A_{total}'(x)=2-\frac{1672}{x^{2}}$.

Step4: Set the derivative equal to zero and solve for $x$.

Set $A_{total}'(x) = 0$, then $2-\frac{1672}{x^{2}}=0$. Rearranging gives $2x^{2}=1672$, so $x^{2}=836$, and $x = 2\sqrt{209}$.

Step5: Find the value of $y$.

Substitute $x = 2\sqrt{209}$ into $y=\frac{418}{x}$. $y=\frac{418}{2\sqrt{209}}=\sqrt{209}$. Since $\sqrt{209}<2\sqrt{209}$, the shorter edge is $\sqrt{209}$ and the longer edge is $2\sqrt{209}$.

Answer:

$2\sqrt{209}$