eet h be the vertical height of the garden and let A be the total area of the garden and borders. Write the objective function \( \mathrm{A}=\left(\frac{190}{\mathrm{~h}}+4\right)(\mathrm{h}+2) \) (Type an expression.) The interval of interest of the objective function is \( (0, \infty) \). (Simplify your answer. Type your answer in interval notation.) The vertical height of the garden that will minimize the total area is \( \sqrt{95} \mathrm{~m} \). (Type an exact answer, using radicals as needed.) The horizontal width of the garden that will minimize the total area is

To determine the horizontal width of the garden that minimizes the total area, let's break down the given information and the objective function step by step. ### Given: - **Vertical height of the garden**: \( h \) - **Total area including borders**: \[ \mathrm{A} = \left(\frac{190}{h} + 4\right)(h + 2) \] - **Optimal vertical height**: \[ h = \sqrt{95} \, \text{m} \] ### Objective: Find the **horizontal width** of the garden (\( w \)) that minimizes the total area \( \mathrm{A} \). ### Steps: 1. **Understand the Objective Function**: - The term \( \frac{190}{h} \) represents the **horizontal width of the garden** (\( w \)), since the garden area \( w \times h = 190 \). - The total width including borders is \( w + 4 \), and the total height including borders is \( h + 2 \). 2. **Expressing the Horizontal Width**: \[ \mathrm{A} = \left(\frac{190}{h} + 4\right)(h + 2) \] Here, \( \frac{190}{h} \) is the garden's width \( w \). 3. **Calculate \( w \) at Optimal \( h \)**: \[ w = \frac{190}{h} \] Substituting \( h = \sqrt{95} \): \[ w = \frac{190}{\sqrt{95}} = \frac{190}{\sqrt{95}} \times \frac{\sqrt{95}}{\sqrt{95}} = \frac{190\sqrt{95}}{95} = 2\sqrt{95} \, \text{m} \] ### Final Answer: The horizontal width of the garden that minimizes the total area is: \[ 2\sqrt{95} \, \text{m} \]