Problem 47 - Algebra 1

Question

Write an equation or an inequality to model the real-life situation. The area \(A\) of a trapezoid is equal to one half times the sum of seven and nine, times a number \(h\) plus seven.

Step-by-Step Solution

Answer

The equation derived from the given real-life situation is \(A = 0.5 * (7 + 9) * (h + 7)\).

Step 1: Identify the known quantities

The area 'A' of the trapezoid, the bases of the trapezoid are seven units and nine units, and an unrealized height 'h'.

Step 2: Real-life situation to formula translation

Translate the given English statement into a mathematical formula. The area of a trapezoid is given by \(A = 0.5 * (b1 + b2) * h\), where 'b1' and 'b2' are the lengths of the bases of the trapezoid and 'h' is the height of the trapezoid. From the statement, we infer that 'b1' is 7, 'b2' is 9, and 'h' is implicitly 'h + 7'.

Step 3: Write the equation

Substitute the known quantities into the formula to form an equation. Therefore, \(A = 0.5 * (7 + 9) * (h + 7)\).

Key Concepts

Trapezoid Area Formula

The trapezoid area formula is a fundamental concept in geometry that allows us to calculate the space inside a trapezoid. A trapezoid, also known as a trapezium in some countries, is a four-sided figure with at least one pair of parallel sides. The formula for the area of a trapezoid is:\[ A = 0.5 imes (b1 + b2) imes h \]- Where: - \(A\) is the area. - \(b1\) and \(b2\) are the lengths of the two parallel sides (bases). - \(h\) is the height, which is the perpendicular distance between the two bases.
Understanding this formula is crucial because it forms the basis for solving problems involving real-life situations with trapezoidal shapes. The formula helps in finding how much space or material you need or have, based on the dimensions given.

Mathematical Modeling

Mathematical modeling involves translating real-world situations into mathematical forms so they can be analyzed and solved. This process often starts by understanding the context and identifying the relevant variables and relationships.
In the case of our trapezoid problem, the real-life situation involves finding the area of a trapezoid given specific dimensions. The bases were seven and nine units, while the height involves the expression \(h + 7\).
By using the area formula for a trapezoid, we'll take the real-life description and convert it into a mathematical equation. This is referred to as a model. Mathematical modeling in this context means:

• Identifying what the real-world problem is asking for (area calculation).
• Choosing the correct mathematical tools or formulas needed (trapezoid area formula).
• Substituting real-world data into these formulas (placing known base lengths and height adjustment into the equation).

This careful translation ensures that the solution derived is accurate and applicable to the real world.

Algebraic Equations

Algebraic equations are at the heart of solving many mathematical problems, especially those involving geometrical figures like a trapezoid. These equations use symbols to represent numbers and variables, allowing us to form relationships and solve for unknowns.
In the trapezoid area problem, once we've established that the formula for the area is \(A = 0.5 \times (7 + 9) \times (h + 7)\), we've formed an algebraic equation.
- This equation allows us to: - Calculate the area when the height \(h\) is known. - Solve for \(h\) if the area is given.
Algebraic equations like this are powerful because they don't just provide one answer. They allow us the flexibility to solve for different variables depending on the information available. Algebra uses operations such as addition, multiplication, and others in forming and solving these expressions.