Problem 57 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. All numbers are accurate to at least two significant digits. In remodeling a house, an architect finds that by adding the same amount to each dimension of a 12 -ft by 16 -ft rectangular room, the area would be increased by \(80 \mathrm{ft}^{2}\). How much must be added to each dimension?

Step-by-Step Solution

Answer

Approximately 2.61 feet must be added to each dimension.

Step 1: Define Variables

Let's denote the amount to be added to each dimension as \(x\). This makes the new dimensions of the room \((12 + x)\) feet by \((16 + x)\) feet.

Step 2: Calculate New Area

The new area of the room after adding \(x\) to each dimension will be given by the equation \((12 + x)(16 + x)\).

Step 3: Set Equation for Area Increase

According to the problem, the area increase is 80 square feet. Set up the equation: \((12 + x)(16 + x) = 192 + 80\), where 192 is the original area (\(12 \times 16\)). So, \((12 + x)(16 + x) = 272\).

Step 4: Expand and Rearrange

Expand the expression on the left side: \(192 + 16x + 12x + x^2 = 272\). Simplify it to get \(x^2 + 28x + 192 - 272 = 0\). Therefore, \(x^2 + 28x - 80 = 0\).

Step 5: Solve the Quadratic Equation

This is a standard quadratic equation \(x^2 + 28x - 80 = 0\). To solve it, use the quadratic formula: \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\), where \(a = 1\), \(b = 28\), and \(c = -80\).

Step 6: Calculate the Discriminant

Calculate the discriminant: \(b^2 - 4ac = 28^2 - 4(1)(-80) = 784 + 320 = 1104\).

Step 7: Apply the Quadratic Formula

Plug the values into the quadratic formula: \(x = \frac{{-28 \pm \sqrt{1104}}}{2}\).

Step 8: Simplify the Solutions

First calculate \(\sqrt{1104} \approx 33.22\). Thus the solutions are: \(x = \frac{{-28 + 33.22}}{2} \approx 2.61\) and \(x = \frac{{-28 - 33.22}}{2} \approx -30.61\). Since \(x\) must be positive, \(x \approx 2.61\).

Key Concepts

Area Calculation

When tasked with expanding the size of a room, calculating the new area plays a crucial role. Initially, the room's dimensions are 12 feet by 16 feet, giving an existing area of \(192 \text{ ft}^2\).
To find how much space would be added by increasing each side by the same amount, we use the formula for the area of a rectangle.

• The original dimensions: \(12 \text{ ft} \times 16 \text{ ft} = 192 \text{ ft}^2\)
• The new dimensions will be \((12 + x) \times (16 + x)\).
• This equation shows how much the area increases as \(x\) increases.

By establishing an area change equation \((12 + x)(16 + x) = 272 \text{ ft}^2\), we understand that the additional space gained must be 80 square feet. Here, the shedding of complex operations into manageable parts is essential for clarity.
This type of calculation often serves as a foundation in remodeling projects, empowering designers to visualize spatial expansions clearly.

Equation Solving

Solving quadratic equations is key in determining unknowns in a variety of contexts, such as our room expansion problem.
We begin by identifying the quadratic nature of the expression:

• After expanding \((12 + x)(16 + x) = 272\), the equation becomes \(x^2 + 28x + 192 = 272\).
• Rearranging gives us \(x^2 + 28x - 80 = 0\).

Quadratic equations have a standard form of \(ax^2 + bx + c = 0\), which here has coefficients \(a = 1\), \(b = 28\), and \(c = -80\).
Using the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\), which helps in finding solutions:

• Calculate the discriminant: \(b^2 - 4ac = 1104\).
• This yields two potential solutions.
• Since dimensions can't be negative, only positive roots are valid: \(x \approx 2.61\).

Understanding the method to solve these equations unveils the potential field applications in design, allowing adept interpretation of algebraic solutions in numeric forms.

Architectural Remodeling

Architectural remodeling frequently relies on mathematical solutions to enhance spatial efficiency and aesthetic appeal. By increasing a room's dimensions, an architect can maximize room utility without compromising structural integrity. In this context, we focused on a simple rectangle.
This process involves:

• Determining the current space and assessing growth potential.
• Using algebraic expressions to calculate how much each dimension can increase.
• Designing by calculating how added dimensions will affect overall space and usability, often guided by aesthetic goals and functional needs.

In this exercise, identifying that adding approximately 2.61 feet to each side achieves the space increase reveals the real-world utility of quadratic equations in making informed architectural decisions.
Approaching these problems produces a robust framework for thoughtful expansion and renovation, ultimately catering to both practical and creative aspirations in modern architecture.