Writing and Solving the Inequality for the Length

Oct 13, 2025 by recoveryshake.com 139 views

In this problem, we are given the width of a rectangle as 40 centimeters and the perimeter as at most 520 centimeters. We need to find the length of the rectangle and then calculate its area. To start, let's recall the formula for the perimeter of a rectangle, which is given by:

P = 2(l + w)

where P is the perimeter, l is the length, and w is the width.

Given that the width is 40 centimeters and the perimeter is at most 520 centimeters, we can set up the following inequality:

2(40) + 2l ≤ 520

Simplifying the inequality, we get:

80 + 2l ≤ 520

Subtracting 80 from both sides, we get:

2l ≤ 440

Dividing both sides by 2, we get:

l ≤ 220

This means that the length of the rectangle is at most 220 centimeters.

Writing and Solving the Inequality for the Length

Let's write the inequality in a more formal way:

l ≤ 220

This inequality tells us that the length of the rectangle is less than or equal to 220 centimeters.

Writing the Inequality for the Area

To find the area of the rectangle, we need to multiply the length and width:

A = l × w

Substituting the values, we get:

A = l × 40

Since we know that the length is at most 220 centimeters, we can write the inequality for the area as:

A ≤ 220 × 40

Simplifying the inequality, we get:

A ≤ 8800

This means that the area of the rectangle is at most 8800 square centimeters.

Conclusion

In this problem, we set up an inequality to find the length of the rectangle and then used that inequality to find the area. We found that the length of the rectangle is at most 220 centimeters and the area is at most 8800 square centimeters.

Key Takeaways

The perimeter of a rectangle is given by the formula P = 2(l + w).
The inequality for the length of the rectangle is l ≤ 220.
The inequality for the area of the rectangle is A ≤ 8800.

Practice Problems

Find the length of a rectangle with a width of 30 centimeters and a perimeter of at most 600 centimeters.
Find the area of a rectangle with a length of 150 centimeters and a width of 20 centimeters.

Solutions

To find the length of the rectangle, we can set up the inequality:

2(30) + 2l ≤ 600

Simplifying the inequality, we get:

60 + 2l ≤ 600

Subtracting 60 from both sides, we get:

2l ≤ 540

Dividing both sides by 2, we get:

l ≤ 270

This means that the length of the rectangle is at most 270 centimeters.

To find the area of the rectangle, we can multiply the length and width:

A = 150 × 20

Simplifying the inequality, we get:

A = 3000

Q: What is the formula for the perimeter of a rectangle?

A: The formula for the perimeter of a rectangle is given by:

P = 2(l + w)

where P is the perimeter, l is the length, and w is the width.

Q: How do I set up an inequality to find the length of a rectangle?

A: To set up an inequality to find the length of a rectangle, you need to know the width and the maximum perimeter. You can use the formula for the perimeter and set up an inequality like this:

2w + 2l ≤ P

where w is the width, l is the length, and P is the maximum perimeter.

Q: How do I solve an inequality to find the length of a rectangle?

A: To solve an inequality to find the length of a rectangle, you need to isolate the variable l. You can do this by subtracting 2w from both sides of the inequality and then dividing both sides by 2.

Q: What is the relationship between the length and width of a rectangle?

A: The length and width of a rectangle are related by the formula for the perimeter. If you know the width and the maximum perimeter, you can use the formula to find the length.

Q: How do I find the area of a rectangle?

A: To find the area of a rectangle, you need to multiply the length and width:

A = l × w

Q: What is the formula for the area of a rectangle?

A: The formula for the area of a rectangle is given by:

A = l × w

where A is the area, l is the length, and w is the width.

Q: Can I use the same inequality to find the area of a rectangle?

A: No, you cannot use the same inequality to find the area of a rectangle. The inequality for the length is:

l ≤ 220

But the area is a product of the length and width, so you need to multiply the length and width to find the area.

Q: How do I find the maximum area of a rectangle?

A: To find the maximum area of a rectangle, you need to find the maximum length and width. You can do this by using the inequality for the length and the formula for the area.

Q: Can I use the same inequality to find the maximum area of a rectangle?

A: No, you cannot use the same inequality to find the maximum area of a rectangle. The inequality for the length is:

l ≤ 220

But the area is a product of the length and width, so you need to multiply the length and width to find the area.

Q: What is the relationship between the perimeter and area of a rectangle?

A: The perimeter and area of a rectangle are related by the formula for the perimeter and the formula for the area. If you know the width and the maximum perimeter, you can use the formula to find the length and then multiply the length and width to find the area.

Q: Can I use the same inequality to find the perimeter and area of a rectangle?

A: No, you cannot use the same inequality to find the perimeter and area of a rectangle. The inequality for the perimeter is:

2w + 2l ≤ P

But the area is a product of the length and width, so you need to multiply the length and width to find the area.

Conclusion

In this article, we have answered some frequently asked questions about finding the length and area of a rectangle. We have also discussed the relationship between the perimeter and area of a rectangle. We hope that this article has been helpful in understanding the concepts of perimeter and area of a rectangle.