The problem in the image involves a square made from thin wire, with a perimeter of 60 cm, and asks three questions:

1. (a) The area of the square.
2. (b) The area of a rectangle made with the wire, where the length is twice its width.
3. (c) The number of smaller squares that can be made from the wire, each with an area of 4 square centimeters.

Let's solve each part.

Part (a) - Area of the Square

The perimeter of the square is given as 60 cm. The perimeter $P$ of a square is calculated by:

$P = 4 \times \text{side length} \quad \text{or} \quad s = \frac{P}{4}$

Substitute the given perimeter:

$s = \frac{60}{4} = 15 \text{ cm}$

The area $A$ of a square is:

$A = s^2 = 15^2 = 225 \text{ cm}^2$

So, the area of the square is 225 cm².

Part (b) - Area of the Rectangle

The length of the rectangle is twice its width. Let's denote:

• $w$ = width
• $2w$ = length

The perimeter $P$ of a rectangle is:

$P = 2(\text{length} + \text{width}) = 2(2w + w) = 6w$

Since the wire is 60 cm long (the perimeter of the rectangle):

$6w = 60 \quad \Rightarrow \quad w = 10 \text{ cm}$

So, the width is 10 cm and the length is:

$2w = 2 \times 10 = 20 \text{ cm}$

The area $A_{\text{rect}}$ of the rectangle is:

$A_{\text{rect}} = \text{length} \times \text{width} = 20 \times 10 = 200 \text{ cm}^2$

So, the area of the rectangle is 200 cm².

Part (c) - Number of Smaller Squares

Each smaller square has an area of 4 cm². The area of each small square is related to its side length $s_{\text{small}}$ by:

$s_{\text{small}} = \sqrt{4} = 2 \text{ cm}$

We need to find how many smaller squares can be made from the total wire area. The total area of the square formed from the wire is 225 cm² (from part (a)).

Now, divide the total area by the area of each small square:

$\text{Number of small squares} = \frac{\text{Total area of wire square}}{\text{Area of each small square}} = \frac{225}{4} = 56.25$

Since we can only have whole squares, we round down to 56 smaller squares.

Would you like more details on any part of this solution?

Related Questions:

1. What is the relationship between the perimeter and area of a square?
2. How does changing the dimensions of a rectangle while keeping the perimeter constant affect its area?
3. What is the formula for calculating the perimeter of any rectangle or square?
4. How do you calculate the area of a square or rectangle given the side lengths?
5. How can you determine the number of smaller shapes that can fit into a larger one?

Tip:

When calculating areas, always make sure to double-check the units and ensure you're consistent throughout the problem!