Problem 42 - Algebra 1: Concepts and Skills

Question

In Exercises 42 and 43, a triangular sign has a base that is 2 feet less than twice its height. A local zoning ordinance restricts the surface area of street signs to be no more than 20 square feet. Write an inequality involving the height that represents the largest triangular sign allowed.

Step-by-Step Solution

Answer

The inequality that represents the largest triangular sign allowed by the ordinance in terms of the height is \(h^2 - h - 20 \leq 0\).

Step 1: Determine the formula for base

The base of the triangle is '2 feet less than twice its height'. This can be translated into the mathematical formula `b = 2h - 2` where `b` is the base and `h` is the height.

Step 2: Plug base formula into the area formula

The area of a triangle formula is `A = 1/2 * b * h` which gives us `A = 1/2 * (2h - 2) * h`.

Step 3: Simplify above equation

The simplified equation yields `A = h^2 - h`.

Step 4: Using the zoning restrictions set up an inequality

Since the area 'A' should not be more than 20 square feet we can create the inequality `h^2 - h <= 20`

Step 5: Rearrange the inequality to standard quadratic form

Rearranging the terms we get `h^2 - h - 20 <= 0`.

Key Concepts

Quadratic Inequalities

When we encounter expressions like \( h^2 - h - 20 \) and are asked to find the range of values for which they hold true under a certain condition, we are dealing with quadratic inequalities. These are similar to quadratic equations, but instead of an equals sign, they feature an inequality: either \( < \) , \( > \) , \( \leq \) , or \( \geq \).

To solve a quadratic inequality:

• Rearrange the inequality in the standard form, \( ax^2 + bx + c \leq 0 \)
• Find the roots of the corresponding quadratic equation, \( ax^2 + bx + c = 0 \)
• Plot these roots on a number line and determine the intervals that satisfy the inequality.

Applying these steps to our problem, \( h^2 - h - 20 \leq 0 \), we need to determine the values of the height \( h \) that result in an area meeting the local zoning ordinance's restriction, thereby finding the largest possible triangular sign. A visual representation on a number line will also aid in understanding the range of heights allowed.

Triangle Area Formula

The area of a triangle is a critical measure in many geometric problems and is calculated using the formula \( A = \frac{1}{2} \times b \times h \), where \( A \) stands for area, \( b \) for base, and \( h \) for height. This formula is derived by considering a triangle as half of a rectangle when the base and height are perpendicular to each other.

In our textbook example, the triangular sign's area must abide by local regulations, not exceeding 20 square feet. Using the expression for the base \( b = 2h - 2 \) and plugging it into our area formula, we express the area solely in terms of the height. This single-variable representation \( A = h^2 - h \) is what we use to establish the inequality for the zoning ordinance's requirement. Understanding this formula is crucial not only for solving geometrical problems but also for understanding how constraints like zoning ordinances mathematically impact design parameters.

Algebraic Expressions

Algebraic expressions are the backbone of algebra and are composed of numbers, variables, and arithmetic operations. In our problem, \( b = 2h - 2 \) is an example of an algebraic expression connecting the base \( b \) of the triangle with its height \( h \).

• Variables like \( b \) and \( h \) represent unknown quantities that can change.
• Constants are the fixed numbers, like the 2 in our expression, signifying a specific number of feet.
• Operators are the plus \( + \) and minus \( - \) signs, indicating addition and subtraction, respectively.

Understanding how to manipulate these expressions—by combining like terms or using the distributive property, for instance—allows us to model and solve real-world problems, like finding the maximum size of a street sign in compliance with local laws. Hence, a firm grasp of algebraic expressions aids in interpreting and setting up inequalities that appear in various practical scenarios.