Given that the length of a rectangle is four times its width and the perimeter is 100 meters, let's find the area of the rectangle.

Step 1: Define Variables

• Let the width of the rectangle be $w$.
• Then the length $l$ is $4w$ (since the length is four times the width).

Step 2: Use the Perimeter Formula

The perimeter $P$ of a rectangle is given by: $P = 2l + 2w$ Substitute the given perimeter and the expression for the length: $100 = 2(4w) + 2w$

Step 3: Solve for $w$

Simplify the equation: $100 = 8w + 2w$ $100 = 10w$ $w = \frac{100}{10} = 10 \text{ meters}$

Step 4: Find the Length

Since $l = 4w$: $l = 4(10) = 40 \text{ meters}$

Step 5: Calculate the Area

The area $A$ of the rectangle is given by: $A = l \times w = 40 \times 10 = 400 \text{ square meters}$

Final Answer:

The area of the rectangle is 400 square meters.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What would the area be if the perimeter were increased to 120 meters?
2. How does changing the ratio between the length and width affect the area?
3. If the width were doubled, what would the new perimeter be?
4. Can you derive a general formula for the area in terms of the perimeter and the ratio between length and width?
5. How would you calculate the diagonal of the rectangle?

Tip: Remember that knowing the relationship between different dimensions (like length and width) can simplify solving for other properties of geometric figures.