Problem 61 - Elementary Algebra

Question

The perimeter of a rectangle is given by the formula \(P=2 l+2 w\), where \(/\) represents the length and \(w\) represents the width. What is the perimeter of a fenced-in rectangular yard measuring 70 feet by 100 feet?

Step-by-Step Solution

Answer

The perimeter is 340 feet.

Step 1: Understand the Formula for Perimeter

The formula for the perimeter of a rectangle is given by:\[ P = 2l + 2w \] where \( l \) is the length and \( w \) is the width of the rectangle.

Step 2: Identify the Given Dimensions

From the problem, we know that the length \( l \) of the rectangular yard is 100 feet, and the width \( w \) is 70 feet.

Step 3: Substitute the Given Values into the Formula

Using the formula \( P = 2l + 2w \), substitute \( l = 100 \) and \( w = 70 \):\[ P = 2(100) + 2(70) \]

Step 4: Calculate the Perimeter

Perform the calculations: \( 2 \times 100 = 200 \) and \( 2 \times 70 = 140 \). Add these two results together:\[ P = 200 + 140 = 340 \]

Step 5: State the Result

The perimeter of the rectangular yard is 340 feet.

Key Concepts

Understanding Length

The length of a rectangle is one of its most important dimensions. It measures how long the rectangle is from one side to the other, usually the longer side. Imagine a rectangular yard; the length would be the measurement you take along the longest side when you're running from one end to the other. In our example, the length of the yard is 100 feet. This means if you lined up 100 feet rulers, end to end, they would stretch the entire long side of the yard. It’s crucial to correctly identify and measure the length in problems like these because it helps determine other properties of the rectangle, such as its perimeter. Remember, length is always measured in linear units, like feet or meters, depending on what is given in your geometry problem.

Deciphering Width

Width is the other key dimension of a rectangle, often the shorter side compared to the length. Think of the width as how wide the rectangle is, which can be seen as the distance across the shorter side of the yard. If length is the long road, width is the crossing street. For our rectangular yard, the width is 70 feet. This is when you measure across the shorter edges of the yard.

• To find the perimeter, the width is equally as important as the length, as it contributes to the total distance around the yard.
• Like length, width is also measured in linear units.

Incorrectly identifying width can lead to errors in calculating the perimeter, so always make sure to properly note which is the length and which is the width.

Exploring Rectangular Yard Perimeters

When dealing with geometry problems, particularly with rectangles, it's common to work with a rectangular yard. In context, a yard is typically a simple example to visualize perimeter problems.The perimeter of a yard is the total distance around the yard's boundary. This means if you were to walk around the yard, starting from one corner and returning to the same corner, you would walk exactly the perimeter distance.

• In this exercise, we calculated the perimeter using the formula: \( P = 2l + 2w \), where we multiply both the length and the width by 2 and then add results together.
• For our 100 by 70 feet yard, the perimeter comes out to 340 feet.

Knowing how to compute the perimeter helps you ensure that any fencing around the yard completely encloses the space. It’s a practical application of geometry in real-world scenarios, ensuring every inch of your yard is taken into account.