A rectangular flower garden with an area of $110 \mathrm{~m}^{2}$ is surrounded by a grass border $1 \mathrm{~m}$ wide on two sides and a $2 \mathrm{~m}$ 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 vertical height of the garden that will minimize the total area is $\mathrm{m}$ (Type an exact answer, using radicals as needed.)

Step 1 ：Let's denote the length of the garden as \(x\) (in meters) and the width as \(y\) (in meters). According to the problem, the area of the garden is 110 m², so we have the equation: \(xy = 110\)

Step 2 ：The total area of the garden and the borders is \((x+2)(y+4)\). We want to minimize this area. To do this, we can express \(y\) in terms of \(x\) using equation (1) and then find the derivative of the total area with respect to \(x\).

Step 3 ：From equation (1), we have \(y = \frac{110}{x}\). Substituting this into the total area gives: \(A = (x+2)((\frac{110}{x})+4) = 110 + 4x + \frac{220}{x} + 8\)

Step 4 ：To find the minimum of this function, we take the derivative with respect to \(x\) and set it equal to zero: \(A' = 4 - \frac{220}{x^2} = 0\)

Step 5 ：Solving for \(x\) gives \(x^2 = \frac{220}{4} = 55\), so \(x = \sqrt{55}\).

Step 6 ：Substituting \(x = \sqrt{55}\) back into equation (1) gives \(y = \frac{110}{\sqrt{55}} = \sqrt{55}\).

Step 7 ：\(\boxed{\text{So, the dimensions that minimize the total area are } x = \sqrt{55} \text{ m and } y = \sqrt{55} \text{ m. The vertical height of the garden that will minimize the total area is } \sqrt{55} \text{ m.}}\)