Problem 43 - 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. Find the base and height of the largest triangular sign that meets the zoning ordinance.

Step-by-Step Solution

Answer

The largest allowed dimensions for the triangular sign under the local zoning ordinance would be \(height \approx 5.47\) feet and \(base \approx 8.94\) feet.

Step 1: Set up the equation

First, the base is defined in relation to the height, so we can write the base as: \(base = 2*height - 2\). Next, we know the area of a triangle is provided by the formula: \(Area = \frac{1}{2} * base * height\). Since the area is not allowed to exceed 20 square feet, we substitute above formulaes to state: \(20 = \frac{1}{2} * (2*height - 2) * height\). Solving this equation will provide us the permissible height.

Step 2: Solve for height

First we simplify the equation to be: \(20 = height * (height - 1)\). This forms a quadratic equation \(0 = height^2 - height - 20\). Now we use the quadratic formula for this equation \(height = \frac{-(-1) ± sqrt{(-1)^2 - 4*1*(-20)}}{2*1}\) which gives us two roots. Since height can't be negative, we take only the positive root.

Step 3: Solve for base

Substitute the positive root from previous step (denoted as h) into the expression for base: \(base = 2*h - 2\). This gives us the largest possible base length for the triangular sign.

Key Concepts

Area of a Triangle

Understanding how to calculate the area of a triangle is critical for solving many geometric problems. Typically, the area of a triangle is found using the formula \(A = \frac{1}{2} \times base \times height\). The 'base' refers to the length of one side of the triangle, and 'height' is the perpendicular distance from that base to the opposite vertex.

In our given exercise, the area of the sign, which is a triangle, must not exceed 20 square feet according to local zoning laws. By knowing the maximum area and the relationship between the sign's base and height, we can form a quadratic equation to solve for the triangle's dimensions. This demonstrates the practical application of geometry in real-world situations, like designing street signs that comply with local ordinances.

Quadratic Formula

When solving quadratic equations, the quadratic formula is a reliable method. It states that the roots of any quadratic equation \(ax^2 + bx + c = 0\) can be calculated using \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

For our exercise, the quadratic equation derived from the area expression is \(height^2 - height - 20 = 0\). Note that the 'a' in the formula corresponds to the coefficient of \(height^2\), 'b' to the coefficient of the 'height' term, and 'c' to the constant term. Applying the quadratic formula helps us find the height of the triangle, from which we can determine the base and ensure the triangular sign is within the legal size limit. The process of using the quadratic formula concretely illustrates how abstract algebraic concepts are routinely used in solving tangible problems.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators that define a specific mathematical relationship. In this scenario, we use the expression for the base of the triangular sign \(base = 2 \times height - 2\) to relate the base to its height.

Expressions allow us to create equations that describe real-world situations. By manipulating these expressions algebraically, as we do when we calculate the permissible area of the triangle or solve the quadratic equation to find the height, we unveil the unknown variables. This exercise highlights the practical use of mathematical expressions in solving for unknown dimensions within prescribed limitations.

Inequalities in Algebra

In math, inequalities are statements about the relative size or order of two values. Algebraic inequalities are used to express a range of possible solutions, as opposed to the definite solutions often provided by equations.

Our exercise doesn't explicitly include an inequality, but the concept underlies the problem because the area of the triangular sign must be 'no more than' 20 square feet. This translates to an inequality \(Area \leq 20\). When solving real-life problems, such as determining the dimensions of a sign to comply with regulations, the use of inequalities allows us to understand and work within such constraints. It teaches us the importance of adhering to given boundaries in problem-solving situations.