Question

Geometry A billboard is 10 feet longer than it is high (see figure). The billboard has 336 square feet of advertising space. What are the dimensions of the billboard?

Short answer

The billboard is approximately 16 feet high and 26 feet long.

Step-by-step solution

Set Up The Equations

Based on what is given, two formulas form for Length (L) and Height (H) which are: \(L = H + 10\) (since length is 10 feet more than the height) and \(L * H = 336\) (since the area is the product of length and height).

Substitute one equation into another

Replace L in the area equation with the relation we got from the first equation: \((H + 10) * H = 336\). Expanding this gives quadratic equation \(H^2 + 10H - 336 = 0\).

Solve the quadratic equation

The quadratic formula (which is \(-b \pm \sqrt{b^2 - 4ac} / 2a)\) is required to solve this equation. Here, \(a = 1\), \(b = 10\), \(c = -336\). Plugging these values into the formula gives \(H \approx 16\) (neglecting the negative value since the height cannot be negative), then \(L = H + 10 \approx 26\).

Key concepts

Solving Quadratic Equations

When encountering algebraic problems involving shapes and sizes, such as finding the dimensions of a billboard, quadratic equations often play a crucial role. A quadratic equation is one that can be formulated in the standard form of \( ax^2 + bx + c = 0 \), where 'a', 'b', and 'c' are constants, and 'x' represents an unknown variable that we aim to solve for.

In the given problem, establishing the relationship between the length and height of the billboard and then applying the information about the area leads us to a quadratic equation in terms of height (H). To solve this equation, \( H^2 + 10H - 336 = 0 \), we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula provides the values of 'x' that satisfy the quadratic equation, which in most physical problems, is limited to the positive solution.

After substituting the values of 'a', 'b', and 'c' from the problem into the quadratic formula, we deduce the height (H) and consequently the length (L) of the billboard. Understanding how to frame a problem into a quadratic equation and then solve it using the quadratic formula is a fundamental skill in algebraic problem-solving.

Algebraic Problem-Solving

Algebraic problem-solving is about making connections between mathematical expressions and real-world scenarios. In our case, the relationship between the length and height of a billboard, and its area. The key is to express the known and unknown quantities in algebraic terms.

Improvement advice often highlights the need for a systematic approach: start by defining variables, write down what you know in the form of equations, and then manipulate these equations to find your unknowns. For example, if a billboard is 10 feet longer than it is high, we express this algebraically as \( L = H + 10 \).

Identifying Relationships

To solve the problem efficiently, it's crucial to identify the relationships among the given pieces of information. The area of the billboard, which is 336 square feet, establishes a connection between the length and height—this is our second equation, \( L \times H = 336 \).

With these equations in hand, we implement algebraic techniques such as substitution to reduce the system of equations to a single variable, which leads us directly into the realm of solving quadratic equations, as seen in this exercise.

Systems of Equations

When multiple variables and their corresponding equations are involved, we are dealing with a system of equations. These systems are omnipresent in algebra and provide a structured framework to tackle multidimensional problems.

In the context of the billboard dimensions, we have a system comprising two equations: one that defines the relationship between length and height \( L = H + 10 \), and another that represents the area \( L \times H = 336 \). To find a solution, we must isolate one variable and substitute it into the other equation, thus reducing our system to a single equation that can be solved conventionally.

Solving via Substitution

An effective method to solve a system of linear equations, like the one we derived for the billboard's size, is substitution. We first express one variable in terms of another, then replace it in the second equation. This generates a quadratic equation, signaling the need to use strategies specific to quadratic solutions, as previously discussed.

This process of substitution not only simplifies the problem but also reinforces the interconnected nature of algebraic concepts. Mastering how to solve systems of equations is vital for students to progress in their mathematical education and apply these skills to situations that model the real world.