Problem 22 - Algebra for College Students

Question

Write a formula that describes the function. See Examples 2 and 3. Express the perimeter \(P\) of a rectangle with width 10 ft as a function of its length \(L\)

Step-by-Step Solution

Answer

\[P(L) = 2L + 20\]

Step 1 - Understand the Perimeter Formula of a Rectangle

The formula for the perimeter of a rectangle is given by the sum of all its sides. Specifically, the perimeter is calculated as: \[P = 2L + 2W\]

Step 2 - Plug in the Given Width

In this problem, the width \(W\) is given as 10 feet. Substitute \(W = 10\) into the perimeter formula: \[P = 2L + 2(10)\]

Step 3 - Simplify the Equation

Simplify the equation from the previous step to describe \(P\) as a function of \(L\): \[P = 2L + 20\]

Step 4 - Final Formula

Express the final simplified equation that describes the perimeter in terms of the length \(L\) of the rectangle: \[P(L) = 2L + 20\]

Key Concepts

Rectangle Geometry

A rectangle is a fundamental shape in geometry. It has four sides, with opposite sides being equal in length. The primary properties to understand about rectangles include:
• Opposite sides are parallel and equal in length.
• All interior angles are right angles (90 degrees).
In the context of our exercise, we are given the width of the rectangle as 10 feet. The task involves expressing the perimeter of the rectangle as a function of its length.

Function Notation

Function notation is a way to represent how one quantity depends on another. It often uses a letter to denote the function and variables to show the input and output. For example, in our exercise, we have a function that represents the perimeter of the rectangle. We write this function as:
P(L) = 2L + 20
Here, P(L) is the perimeter function in terms of the rectangle's length (L). Function notation simplifies the expression of mathematical relationships and makes it easier to manipulate equations.

Algebraic Simplification

Algebraic simplification involves manipulating an equation to make it as simple as possible. This includes combining like terms and reducing fractions. In our exercise, we started with the perimeter formula:
P = 2L + 2W
Then, we substituted the given width (W = 10 feet):
P = 2L + 2(10)
Afterward, we simplified it to:
P = 2L + 20
This final simplified equation allows us to clearly see how the perimeter changes with the rectangle's length. This process is essential in algebra to make equations easier to understand and solve.