Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem analytically, and support your solution graphically. Dimensions of a Square If the length of a side of a square is increased by 3 centimeters, the perimeter of the new square is 40 centimeters more than twice the length of the side of the original square. Find the length of the side of the original square.

Step-by-Step Solution

Answer

The length of the side of the original square is 14 cm.

Step 1: Define the Variables

Let \( x \) be the length of the side of the original square in centimeters. The new side length when increased by 3 cm is \( x + 3 \) cm.

Step 2: Express the Perimeter of the New Square

The perimeter of a square is calculated as four times the side length. For the new square, the perimeter is \( 4(x + 3) \).

Step 3: Set Up the Equation

We know that the perimeter of the new square is 40 cm more than twice the original side length. Thus, we set up the equation: \[ 4(x + 3) = 2x + 40 \]

Step 4: Solve the Equation for x

Expand and simplify the equation:\[ 4(x + 3) = 2x + 40 \] \[ 4x + 12 = 2x + 40 \]Subtract \( 2x \) from both sides: \[ 2x + 12 = 40 \] Subtract 12 from both sides: \[ 2x = 28 \] Divide both sides by 2: \[ x = 14 \]

Step 5: Graphical Interpretation

Plot graph of the equation in step 3 to visually verify the solution. Both sides of the equation should intersect at \( x = 14 \). The y-axis represents the value of expressions, and the x-axis represents the length of side \( x \) in centimeters.

Key Concepts

Analytical Methods

Analytical methods are powerful tools used in mathematics to solve problems using a logical approach. They involve breaking down a problem step-by-step and solving it through algebraic manipulation.
In our exercise, we're tasked with finding the length of a side of a square. Using an analytical method, we start by defining variables. Letting \( x \) represent the side of the original square is crucial.
Next, we develop a clear equation that represents the problem's conditions. Analytical methods guide us to a solution by systematically manipulating this equation. This approach helps in understanding the relationship between the different components of the problem, such as the perimeter and the modified side length.
By practicing such methods, you're able to develop robust problem-solving skills that apply to a range of mathematical problems beyond squares and geometry.

Equations

Equations form the backbone of solving mathematical problems analytically. They represent relationships between various quantities, often expressed in terms of variables. In our case, the critical equation we used was:
\[ 4(x + 3) = 2x + 40 \]
This equation captures the relationship between the increment in the side of the square and the resulting change in the perimeter.

• The expression \( 4(x + 3) \) calculates the perimeter of the new square.
• \( 2x + 40 \) represents twice the original side of the square plus an additional 40cm as described in the problem statement.

Solving this equation step-by-step involves algebraic skills like expanding and combining like terms, which are fundamental in any precalculus study. Equations not only help find numerical solutions but also illustrate how different quantities relate within a problem.

Graphical Solutions

Graphical solutions provide a visual way to interpret mathematical problems, complementing analytical methods. By graphing equations, we can see where different expressions balance each other out visually.
For this problem, plotting the equation \( 4(x + 3) - (2x + 40) = 0 \) on a graph, both sides of the equation are represented as functions.

• The x-axis represents potential side lengths, \( x \).
• The y-axis shows the difference between \( 4(x + 3) \) and \( 2x + 40 \).

Intersecting at \( x = 14 \) confirms our algebraic solution, showing that the graphical approach can validate the equation-solving process. By leveraging both methods, students enhance their understanding by seeing equations come to life through graphs.

Geometry

Understanding geometric concepts is essential in dissecting and solving mathematical problems involving shapes. Geometry deals with the properties and relations of points, lines, surfaces, and solids.
In this exercise, we are particularly interested in the properties of squares. Squares are always equilateral, meaning all sides are equal. Their perimeter, calculated as \( 4 \times \text{(side length)} \), informs us of their geometric robustness.

• Initially, the side is \( x \), and upon increment, it becomes \( x + 3 \).
• This addition fundamentally affects its geometry and provides a new perimeter to explore.

By linking geometry with algebra through calculations of perimeters and solving resultant equations, learners grasp how geometry isn't just theoretical but applies to real-life scenarios, allowing comprehensive preparation for further mathematical study.