Let's tackle each part of this question systematically:

Part a) Draw a diagram of the situation.

Since I cannot directly draw here, I'll describe what the diagram should look like:

• Start by sketching a square to represent the original playground. Label each side of the square as $s$, where $s$ is the length of each side.
• Then, draw a larger square around the original one, representing the new playground after the dimensions are increased by 5 meters. Each side of this new square will now be $s + 5$ meters.
• Label the dimensions of the larger square as $s + 5$.

Part b) Find a simplified algebraic expression for the area of the new playground.

• The side length of the new square is $s + 5$.
• The area $A_{\text{new}}$ of a square is found using the formula $\text{Area} = \text{side}^2$.

Thus, the area of the new playground is: $A_{\text{new}} = (s + 5)^2$ Expanding this expression: $A_{\text{new}} = s^2 + 10s + 25$

Part c) Find a simplified algebraic expression for the increase in area.

The increase in area is the difference between the area of the new playground and the area of the original playground.

• The area of the original playground is $A_{\text{original}} = s^2$.
• The increase in area $\Delta A$ is: $\Delta A = A_{\text{new}} - A_{\text{original}} = (s^2 + 10s + 25) - s^2$ Simplifying: $\Delta A = 10s + 25$

Summary of answers:

• a) Diagram: A square $s \times s$ expanded to a new square $(s+5) \times (s+5)$.
• b) The area of the new playground: $A_{\text{new}} = s^2 + 10s + 25$.
• c) The increase in area: $\Delta A = 10s + 25$.

Would you like further details on any part?

Here are 5 follow-up questions:

1. How would the result change if the increase was 10 meters instead of 5 meters?
2. How does the increase in area relate to the perimeter of the original square?
3. What would happen to the area if the sides were decreased instead of increased by 5 meters?
4. Can you derive a general formula for the increase in area if the side length is increased by any amount $x$?
5. How would the algebraic expressions change if the original playground were rectangular?

Tip: Whenever you're expanding binomials, use the distributive property carefully: $(a+b)^2 = a^2 + 2ab + b^2$.