Let me solve both questions step by step.

Problem 1: Finding the closest point on $y = \sqrt{x}$ to (3, 0)

Solution Steps:

1. Distance Formula:
   The distance $D$ from a point $(x, y)$ on the curve $y = \sqrt{x}$ to $(3, 0)$ is: $D = \sqrt{(x - 3)^2 + (\sqrt{x})^2}.$

2. Simplify $D^2$ for Optimization:
   Minimize $D^2$ (since minimizing $D^2$ also minimizes $D$): $D^2 = (x - 3)^2 + x.$ Expand: $D^2 = x^2 - 6x + 9 + x = x^2 - 5x + 9.$

3. Find Critical Points:
   Take the derivative of $D^2$ with respect to $x$ and set it to 0: $\frac{d(D^2)}{dx} = 2x - 5 = 0.$ Solve: $x = \frac{5}{2} = 2.5.$

4. Check $y$ Value:
   Since $y = \sqrt{x}$, the point is: $\left(2.5, \sqrt{2.5}\right).$

Final Answer for Problem 1:

The closest point on the curve is: $\left(2.5, \sqrt{2.5}\right).$

Problem 2: Maximizing Area of the Window

Objective Function and Constraints

1. Perimeter Constraint: The total perimeter includes:

   • Rectangle: $2h + 2r$ (height and width).
   • Semicircle: $\pi r$ (half circumference of a full circle).

   Given $600$ feet of material: $2h + 2r + \pi r = 600.$

   Simplify: $h = \frac{600 - (2 + \pi)r}{2}.$

2. Area Function: The total area includes:

   • Rectangle: $2r \cdot h$.
   • Semicircle: $\frac{1}{2} \pi r^2$.

   Total area: $A = 2r h + \frac{1}{2} \pi r^2.$

   Substitute $h$ from the perimeter constraint: $A = 2r \left(\frac{600 - (2 + \pi)r}{2}\right) + \frac{1}{2} \pi r^2.$

   Simplify: $A = r(600 - (2 + \pi)r) + \frac{1}{2} \pi r^2.$ $A = 600r - (2 + \pi)r^2 + \frac{1}{2} \pi r^2.$ Combine terms: $A = 600r - \left(2 + \frac{\pi}{2}\right)r^2.$

Maximization Using Derivatives

1. First Derivative: $\frac{dA}{dr} = 600 - 2\left(2 + \frac{\pi}{2}\right)r.$ Simplify: $\frac{dA}{dr} = 600 - \left(4 + \pi\right)r.$

2. Critical Point: Set $\frac{dA}{dr} = 0$: $600 = (4 + \pi)r.$ Solve for $r$: $r = \frac{600}{4 + \pi}.$

3. Height $h$: Substitute $r$ back into the equation for $h$: $h = \frac{600 - (2 + \pi)r}{2}.$

Final Answer for Problem 2:

The dimensions of the window (radius $r$ and height $h$) can now be calculated numerically. Let me know if you want me to compute these values!

Follow-Up Questions:

1. Would you like me to compute the exact numerical values for $r$ and $h$?
2. Do you want a step-by-step explanation of how I set up the area function?
3. Would you like additional insight into solving optimization problems using derivatives?
4. Are there alternative methods you’d like to explore for these problems?
5. Should I plot or visualize the solutions for better understanding?

Tip:

When solving optimization problems, always verify that the critical point yields a maximum or minimum by testing endpoints or using the second derivative test.