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Chapter 17: Problem 53

Solve the given problems by setting up and solving appropriate inequalities. Graph each solution. A rectangular PV (photovoltaic) solar panel is designed to be \(1.42 \mathrm{m}\) long and supply \(130 \mathrm{W} / \mathrm{m}^{2}\) of power. What must the width of the panel be in order to supply between \(100 \mathrm{W}\) and \(150 \mathrm{W} ?\)

Short Answer

Expert verified
The width must be between 0.542 m and 0.812 m.

Step by step solution

01

Understand the Problem

The problem involves determining the width of a solar panel that supplies between 100 W and 150 W of power. The panel has a fixed length of 1.42 m and power density of 130 W/m虏.
02

Set Up the Power Inequality

Let the width be denoted by \( w \). The area of the panel is \( 1.42 \times w \) square meters. The power supplied can be calculated as \( 130 \times 1.42 \times w \). We need to find values of \( w \) that satisfy \( 100 \leq 130 \times 1.42 \times w \leq 150 \).
03

Simplify the Inequality

Calculate the product of 130 and 1.42, which is 184.6. Substitute this back into the inequality: \( 100 \leq 184.6w \leq 150 \).
04

Solve the Inequality

To find \( w \), divide the entire inequality by 184.6. This gives \( \frac{100}{184.6} \leq w \leq \frac{150}{184.6} \). Compute the division to get approximate values: \( 0.542 \leq w \leq 0.812 \).
05

Graph the Solution

On a number line, draw an interval from approximately 0.542 to 0.812. Shade the region between them to represent all possible widths of the panel that satisfy the power requirement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solar Panel Calculation
The process of calculating the necessary dimensions of a solar panel is crucial to ensure that it delivers the desired amount of electrical power. Solar panels convert sunlight into electricity, and understanding how to determine their required size involves some straightforward computation.
The given problem relates to a solar panel that has a fixed length of 1.42 meters. This panel is designed for a power output between 100 watts and 150 watts based on a power density of 130 watts per square meter.In order to find out the appropriate width that meets this power requirement:
  • First, understand that the area of the solar panel, which affects its power output, depends on its dimensions - specifically, its length and width.
  • The formula for the area of the panel is simply the length multiplied by the width, i.e., area = length \( \times \) width. In this problem, it is expressed as \( 1.42 \times w \), where \( w \) is the width.
  • The power generated by the panel is calculated by multiplying the area with the power density: \( 130 \times 1.42 \times w \). To satisfy the problem's restrictions, this computation must result in a power output between 100 and 150 watts.
This context provides the foundation for setting up the inequality that will help us find the width.
Area and Power Relations
When dealing with solar panels, the relationship between area and power is pivotal. The amount of power a solar panel can generate is directly proportional to its area, assuming a constant power density.
Power density refers to the quantity of power generated per unit area, typically measured in watts per square meter.For this specific problem:
  • The power generated is determined by the product of power density (130 W/m虏) and the panel's area.
  • The area, in turn, is affected by the panel's dimensions, which include a fixed length (1.42 m) and a variable width.
  • In practical terms, increasing the width of the solar panel will increase its area, thereby increasing the power output, as long as the power density remains constant.
This understanding is essential for setting up the inequality that governs the desired power range. Hence, the inequality used in this task (\( 100 \leq 130 \times 1.42 \times w \leq 150 \)) represents the mathematical expression of this relationship, balancing all these factors.
Graphing Solutions of Inequalities
Graphing inequalities is a useful way to visualize solutions in a given context, providing a clear representation of the range of values that satisfy the criteria. In this problem, graphing the solution helps us understand the range of widths the solar panel can have to meet its power output requirements.
A number line is typically used for such visualizations.In practice:
  • The solution to the inequality \( 0.542 \leq w \leq 0.812 \) is graphed by shading the interval from 0.542 to 0.812 on the number line.
  • This shaded region represents all the possible values of the width that will ensure the power output remains within the specified range of 100 to 150 watts.
  • This visual tool not only helps determine feasible solutions but also aids in understanding how altering one parameter (such as width) affects overall outcomes.
Learning to graph solutions in this way is quite beneficial, especially in applied contexts like this, where understanding optimal dimensions is key to designing efficient, functional systems.

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