To determine the two equivalent expressions representing the total fencing material required to surround the playground and flower beds, let's break down the problem.

Step 1: Calculate the Perimeter of the Playground

The playground is a rectangle with:

• Length = $l$
• Width = $w$

The perimeter of the rectangle is given by: $\text{Perimeter of Playground} = 2(l + w)$

Step 2: Calculate the Perimeter of the Flower Beds

Each flower bed is a square with:

• Side length = $a$

The perimeter of one square flower bed is: $\text{Perimeter of one Flower Bed} = 4a$

Since there are 2 square flower beds, the total perimeter for both is: $\text{Perimeter of two Flower Beds} = 2 \times 4a = 8a$

Step 3: Combine the Perimeters

The total fencing material required to surround both the playground and the two flower beds is the sum of their perimeters: $\text{Total Fencing Material} = 2(l + w) + 8a$

Step 4: Simplify the Expression

We can simplify $2(l + w) + 8a$ by distributing the $2$ in the first term: $\text{Total Fencing Material} = 2l + 2w + 8a$

Equivalent Expressions

From the steps above, the two equivalent expressions for the total fencing material are: $2(l + w) + 8a \quad \text{and} \quad 2l + 2w + 8a$

The other expressions you mentioned, such as $2lw + 2alw + 4a$, do not accurately represent the total perimeter, so they are not equivalent to the correct expressions.

Conclusion

The correct equivalent expressions for the total fencing material required to surround the playground and flower beds are:

1. $2(l + w) + 8a$
2. $2l + 2w + 8a$

Would you like more details or have any further questions?

Related Questions:

1. How do you find the area of a rectangular playground?
2. What is the difference between perimeter and area?
3. How do you calculate the perimeter of composite shapes?
4. How do you solve problems involving the fencing of multiple shapes?
5. What would be the effect on the total fencing required if the dimensions of the playground or flower beds were doubled?

Tip: Always draw a diagram for complex word problems. It helps visualize the problem and identify the correct formula to use.