Problem 60 - Basic Technical Mathematics with Calculus

Question

Set up the appropriate quadratic equations and solve. A rectangular solar panel is \(20 \mathrm{cm}\) by \(30 \mathrm{cm} .\) By adding the same amount to each dimension, the area is doubled. How much is added?

Step-by-Step Solution

Answer

Each dimension is increased by 10 cm.

Step 1: Understand the Problem

A rectangular panel is originally 20 cm by 30 cm, resulting in an area of \(20 \times 30 = 600 \text{ cm}^2\). We need to find a length \(x\) which, when added to both dimensions, doubles this area.

Step 2: Set Up New Dimensions

When \(x\) is added to both the length and the width, the dimensions become \((20 + x)\) cm and \((30 + x)\) cm.

Step 3: Express the Doubled Area

Doubling the original area gives: \(2 \times 600 = 1200 \text{ cm}^2\). Hence, the equation for the areas becomes \((20 + x)(30 + x) = 1200\).

Step 4: Formulate the Quadratic Equation

Expand the equation: \((20 + x)(30 + x) = 600 + 40x + 30x + x^2 = 1200\).Combine like terms: \(x^2 + 50x + 600 = 1200\).Rearrange to form the quadratic equation: \(x^2 + 50x + 600 - 1200 = 0\) or \(x^2 + 50x - 600 = 0\).

Step 5: Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = 50\), and \(c = -600\).Calculate \(b^2 - 4ac = 50^2 - 4 \times 1 \times (-600) = 2500 + 2400 = 4900\).Thus, \(x = \frac{-50 \pm \sqrt{4900}}{2} = \frac{-50 \pm 70}{2}\).

Step 6: Determine the Values of x

Calculate the possible values of \(x\):1. \(x = \frac{-50 + 70}{2} = \frac{20}{2} = 10\).2. \(x = \frac{-50 - 70}{2} = \frac{-120}{2} = -60\).Since a dimension cannot be negative, the solution is \(x = 10\).

Key Concepts

Rectangular Dimensions

Understanding rectangular dimensions is crucial when dealing with problems related to geometry, such as the one about the solar panel. A rectangle is defined by its length and width, which are the two dimensions needed to calculate its area. In our problem, the initial dimensions are given as 20 cm and 30 cm.

When we add a variable, let's call it \( x \), to each of these dimensions, we effectively increase both the length and the width by the same amount. This transforms our rectangle into one with new dimensions: \((20 + x)\) cm by \((30 + x)\) cm.

• The term "dimension" essentially refers to measurable extents of an object, which in the case of rectangles, involves length and width.
• When both dimensions increase by a same factor, the overall size, and hence area of the rectangle, changes significantly.
• In practical applications like designing solar panels, understanding how dimensions interact with each other is fundamental to optimizing space and efficiency.

Area Calculation

The area of a rectangle is calculated through the simple formula: length times width, or \( ext{Area} = ext{length} imes ext{width} \). For our original solar panel, with dimensions 20 cm by 30 cm, the area is \(600 \text{ cm}^2\).

When we add \( x \) to each dimension (making the lengths \((20 + x)\) and \((30 + x)\)), you must recalculate the new area using these updated dimensions. The equation \((20 + x)(30 + x) = 1200\) demonstrates that the new area must equal \(1200 \text{ cm}^2\).

• Start with the basic formula and ensure you correctly substitute all values.
• Understanding how changes in dimensions affect area lets you double check your solutions, especially in practical applications where precision is important.
• Area calculation is common in architecture, design, and other fields where space optimization is key.

Quadratic Formula

Once we have set up the problem into a quadratic equation like \( x^2 + 50x - 600 = 0 \), we need a method to find \( x \). Here, the quadratic formula is incredibly useful. It's given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a \), \( b \), and \( c \) come from our equation. For example, \( a = 1 \), \( b = 50 \), and \( c = -600 \).

This reliable method allows us to find both potential solutions, which in this scenario might be realistic (positive) or not practical (negative, because dimensions can't be negative).

• The quadratic formula is a universal tool to solve quadratic equations, providing solutions when factoring is cumbersome or impossible.
• It involves discerning the discriminant \( b^2 - 4ac \), as it determines the nature of the roots (real or complex, distinct or repeated).
• Upon solving, always consider the practicality of your solutions. Values must make sense within the context of the problem.

This technique transcends beyond simple math, and is employed in physics, engineering, and economics, offering solutions to real-world problems where relationships are perfectly quadratic.