Problem 274 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. The perimeter of a rectangular atrium is 160 feet. The length is 16 feet more than the width. Find the length and width of the atrium.

Step-by-Step Solution

Answer

The width is 32 feet and the length is 48 feet.

Step 1: Define Variables

Let the width of the rectangle be denoted as \(w\) and the length be denoted as \(l\).

Step 2: Set Up Length Equation

According to the problem, the length is 16 feet more than the width. Therefore, we can write the equation: \(l = w + 16\).

Step 3: Set Up Perimeter Equation

The perimeter of a rectangle is given by the formula \(P = 2l + 2w\). Given that the perimeter is 160 feet, we set up the equation: \(2l + 2w = 160\).

Step 4: Substitute Length Equation

Substitute the equation for \(l\) into the perimeter equation: \[2(w + 16) + 2w = 160\].

Step 5: Simplify the Equation

Distribute and combine like terms: \[2w + 32 + 2w = 160\] simplifies to \[4w + 32 = 160\].

Step 6: Solve for Width

Subtract 32 from both sides: \[4w = 128\]. Then, divide by 4: \[w = 32\].

Step 7: Solve for Length

Use the relation \(l = w + 16\) to find the length: \[l = 32 + 16 = 48\].

Step 8: Verify the Solution

Check that the values satisfy the original perimeter equation: \[2(48) + 2(32) = 96 + 64 = 160\].

Key Concepts

perimeter of rectangle

The perimeter of a rectangle is the total distance around the outside of the rectangle. It can easily be calculated using the formula:
\[ P = 2l + 2w \]
where \( l \) is the length and \( w \) is the width. In this exercise, we know the perimeter is 160 feet, which helps us create an equation that relates the length and width. By setting up the perimeter equation, we can further solve for either the length or width, provided we have more information.

solving equations

Solving equations is a fundamental algebraic skill. Here, we start with an equation that describes the relationship between the perimeter, length, and width of the rectangle. Initially, we have:

• Length equation: \( l = w + 16 \)
• Perimeter equation: \( 2l + 2w = 160 \)

We substitute the length equation into the perimeter equation to solve for the width \( w \). This substitution transforms our perimeter equation into a single-variable equation
\[ 2(w + 16) + 2w = 160 \]
which can be simplified and solved step-by-step.

algebraic expressions

In this problem, understanding and manipulating algebraic expressions is crucial. An algebraic expression is a combination of variables, numbers, and operations. When we substitute one variable into another equation (as we did with \( l = w + 16 \) into the perimeter equation), we create a new expression:
\[ 4w + 32 = 160 \]After simplifying this, we isolate the variable \( w \) by performing arithmetic operations like subtraction and division. Simplifying algebraic expressions involves combining like terms and ensuring both sides of the equation are balanced, leading us to a solution.

geometry

Geometry helps us understand and solve problems related to shapes and spaces. Rectangles, a basic geometric shape, are defined by their length and width. Knowing the properties of rectangles, such as the perimeter formula, aids in solving real-world problems. By applying geometric concepts, we not only solve for the dimensions but also verify the solution:

• If \( w = 32 \) feet and \( l \) is 16 feet more, then \( l = 48 \) feet.
• Verifying: \( 2(48) + 2(32) = 160 \) feet, confirming our calculations.