Let's solve each problem one by one.

Problem 11:

Problem Statement: The width of a rectangular dance floor is $w$ feet. The length of the floor is 6 feet longer than its width. Which of the following expresses the perimeter, in feet, of the dance floor in terms of $w$?

Solution:

1. Width of the floor: $w$ feet.

2. Length of the floor: Since the length is 6 feet longer than the width, the length is $w + 6$ feet.

3. Perimeter of a rectangle: The formula for the perimeter $P$ of a rectangle is given by: $P = 2 \times (\text{Length} + \text{Width})$ Substituting the given values: $P = 2 \times (w + (w + 6)) = 2 \times (2w + 6)$ Simplify the expression: $P = 4w + 12$

   The correct option is $\boxed{D} \, 4w + 12$.

Problem 12:

Problem Statement: In the figure, a circle is inscribed in a square with an area of 25 square units. What is the area, in square units, of the shaded region?

Solution:

1. Area of the square: The area of the square is given as 25 square units.

2. Side length of the square: Since the area of the square is given by $\text{side}^2$, we find the side length as: $\text{Side length} = \sqrt{25} = 5 \text{ units}$

3. Diameter of the circle: The circle is inscribed in the square, so the diameter of the circle is equal to the side length of the square, which is 5 units.

4. Radius of the circle: The radius $r$ of the circle is: $r = \frac{\text{Diameter}}{2} = \frac{5}{2} = 2.5 \text{ units}$

5. Area of the circle: The area $A$ of the circle is given by: $A_{\text{circle}} = \pi r^2 = \pi (2.5)^2 = 6.25\pi \text{ square units}$

6. Area of the shaded region: The shaded region is the area of the square minus the area of the circle: $A_{\text{shaded}} = \text{Area of square} - \text{Area of circle} = 25 - 6.25\pi \text{ square units}$

   The correct option is $\boxed{B} \, 25 - 6.25\pi$.

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Do you have any questions or need further details? Here are five related questions to consider:

1. How would the perimeter formula change if the length of the rectangle was a different amount longer than the width?
2. What would happen to the area of the shaded region if the square's area was doubled?
3. How do you calculate the perimeter of more complex shapes like trapezoids?
4. What is the relationship between the area and perimeter of a square when the side length is doubled?
5. How does the area of a circle change when its radius is halved?

Tip: Always double-check your final expression by substituting values to see if it aligns with the problem's conditions.