Write an expression to represent the perimeter of the rectangle

1 Expert Answer

Jason L. answered • 09/08/16

Graduate Student Who Loves to Do Math

P = 2l + 2w

l = 3w

Now plug in.

P = 2(3w) + 2w

Note that this further simplifies to 8w, but the answer is C.

A rectangle is a parallelogram with four right angles. All rectangles are also parallelograms, but not all parallelograms are rectangles.

The perimeter P of a rectangle is given by the formula, P = 2 l + 2 w , where l is the length and w is the width of the rectangle.

The area A of a rectangle is given by the formula, A = l w , where l is the length and w is the width.

You will often encounter word problems where two of the values in one of these formulas are given, and you are required to find the third.

Example 1:

The perimeter of a rectangular pool is 56 meters. If the length of the pool is 16 meters, then find its width.

Here the perimeter and the length of the rectangular pool are given. We have to find the width of the pool.

The perimeter P of a rectangle is given by the formula, P = 2 l + 2 w , where l is the length and w is the width of the rectangle.

Given that, the perimeter is 56 meters and the length is 16 meters. So, substitute these values into the formula.

56 = 2 ( 16 ) + 2 w

Simplify.

56 = 32 + 2 w

Subtract 32 from both sides.

24 = 2 w

Divide each side by 2 .

12 = w

Therefore, the width of the rectangular pool is 12 meters.

Example 2:

The area of a rectangular fence is 500 square feet. If the width of the fence is 20 feet, then find its length.

Here the area and the width of the rectangular fence are given. We have to find the length of the fence.

The area A of a rectangle is given by the formula, A = l w , where l is the length and w is the width.

Given that, the area is 500 square feet and the width is 20 feet. So, substitute these values into the formula.

500 = l × 20

Divide each side by 20 to isolate l .

25 = l

Therefore, the length of the rectangular fence is 25 feet.

Writing Algebraic Expressions for the Perimeter of a Figure

Step 1: Identify the side lengths of the given figure.

Step 2: Add the side lengths together to get an algebraic expression representing the perimeter of the given figure.

Step 3: Simplify the algebraic expression by combining like terms where possible.

Writing Algebraic Expressions for the Perimeter of a Figure Vocabulary

Perimeter: The perimeter of a figure is the length of the boundary of the figure. In other words, the perimeter of a figure can be found by adding up the lengths of all of the sides of the figure. For example, if we put a fence up, such that it is a border of a lawn, then the total length of the fence is equivalent to the perimeter of the lawn.

Algebraic expressions: Algebraic expressions are expressions that are a combination of variables and numbers that are connected by arithmetic operations. For example, {eq}2x+5 {/eq} and {eq}(3x-2) \div 4 {/eq} are algebraic expressions.

Using these steps and definitions, we will solve two examples of finding the perimeter of a given figure. The first example will be a figure that has four sides, and the second example will be a figure that has five sides.

Writing Algebraic Expressions for the Perimeter of a Figure: Example 1

Find the perimeter of the given figure QRPO:

Step 1: Identify the side lengths of the given figure.

The side lengths of the given figure are given as {eq}12, (4x + 2), (2x + 8), \text{ and }(5x + 6) {/eq}.

Step 2: Add the side lengths together to get an algebraic expression representing the perimeter of the given figure.

The perimeter is the sum of all the side lengths. Therefore, we add the lengths of the sides together to get an algebraic expression representing the perimeter of our figure.

$$P =12 + (4x + 2) + (2x + 8) + (5x + 6) $$

Step 3: Simplify the algebraic expression by combining like terms where possible.

We can remove the parentheses in our expression since we aren't distributing anything through them. This leaves us with:

$$P =12 + 4x + 2 + 2x + 8 + 5x + 6 $$

Like terms are terms that have the same variables raised to the same exponent, and we add like terms by adding their coefficients and leaving their variable factors as they are. Since 4x, 2x, and 5x all contain an x raised to the first power, they are like terms. Since 12, 2, 8, and 6 do not contain a variable, they are like terms. Thus, we simplify our expression by combining like terms as follows:

$$\begin{align} P &= \color{Red}{12} + \color{Blue}{4x} + \color{Red}{2} + \color{Blue}{2x} +\color{Red}{8} + \color{Blue}{5x} + \color{Red}{6}\\ P &= \color{Red}{(12 + 2 + 8 + 6)} + \color{Blue}{(4 + 2 + 5)x}\\ P &= \color{Red}{28} + \color{Blue}{11x} \end{align} $$

We see that the perimeter of the given figure is {eq}\boxed{\bf{28 + 11x}} {/eq}.

Writing Algebraic Expressions for the Perimeter of a Figure: Example 2

Find the perimeter of the given figure:

Step 1: Identify the side lengths of the given figure.

The side lengths of the given figure are {eq}(2x + 8), 2x, (x + 7), 10, \text{ and }(5x + 4) {/eq}

Step 2: Add the side lengths together to get an algebraic expression representing the perimeter of the given figure.

$$\begin{align} P &=(2x + 8) + 2x + (x + 7) + 10 + (5x + 4)\\ P &=2x + 8 + 2x + x + 7 + 10 + 5x + 4 \end{align} $$

Step 3: Simplify the algebraic expression by combining like terms where possible.

$$\begin{align} P &= \color{Red}{2x} + \color{Blue}{8} + \color{Red}{2x} + \color{Red}{x} +\color{Blue}{7} + \color{Blue}{10} + \color{Red}{5x}+\color{Blue}{4}\\ P &= \color{Red}{(2 + 2 + 1 + 5)x} + \color{Blue}{(8 + 7 + 10 + 4)}\\ P &= \color{Red}{10x} + \color{Blue}{29} \end{align} $$

The perimeter of the given figure is {eq}\boxed{\bf{10x + 29}} {/eq}.

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