Question

A triangle's base is 16 feet less than 2 times its height. If \(h\) represents the height in feet, and the total area of the triangle is 48 square feet, which of the following equations can be used to determine the height? (A) \(2 h+2(h+4)=48\) (B) \(h^{2}-8 h=48\) (c) \(h^{2}+8 h=48\) (D) \(2 h^{2}-16 h=48\)

Short answer

The equation that can be used to determine the height of the triangle is (D) \(2h^2 - 16h = 48\)

Step-by-step solution

Understand the Problem

The problem gives the following information - a triangle's base is 16 feet less than 2 times its height, and the area of the triangle is 48 square feet. The task is to come up with an equation relating the base, height and the area of the triangle.

Infer the Relationship

The area of a triangle is given by the formula - \(A = 0.5 * base * height\). The problem states that the base is 16 feet less than 2 times the height, which can be written as \(base = 2h - 16\). This can be substituted into the formula for the area of the triangle.

Substitute and Simplify

Substituting the value of area and base into the area formula, we get \(48 = 0.5 * (2h - 16) * h\). Simplifying this we get \(2h^2 - 16h - 48 = 0\) which matches with answer choice D.

Key concepts

Algebraic Expressions

Algebraic expressions are the backbone of algebra and involve variables, numbers, and arithmetic operations. They represent relationships between quantities and are foundational in formulating equations. In the context of geometry and specifically triangle area calculations, algebraic expressions are used to symbolize the base, height, and area of the triangle.

For example, if a triangle's base is twice its height minus 16 feet, as stated in the problem, we can express the base in terms of the height variable, 'h', as an algebraic expression: \( base = 2h - 16 \). This allows us to relate the base directly to the height, and when the triangle's area is also given, it enables us to set up an equation to find the unknown height, demonstrating the powerful role algebraic expressions play in solving geometric problems.

Triangle Geometry

Triangle geometry is an essential branch of mathematics that deals with the properties and relationships of triangles. Understanding the characteristics of different types of triangles and the formulas associated with them is key to solving numerous geometrical problems.

The formula for the area of a triangle, \( A = 0.5 \times base \times height \), is fundamental. It incorporates both the base and height of the triangle, which are primary geometric parameters. In our exercise, we're provided with a direct relationship between the base and height, enabling us to set up an equation using these important geometric elements to find out the unknown height of the triangle.

Solving Quadratic Equations

Quadratic equations are a significant part of algebra and can represent a range of real-world problems, including those found in triangle geometry. A quadratic equation is typically in the form \(ax^2 + bx + c = 0\), where 'a', 'b', and 'c' are constants, and 'x' represents the unknown variable.

When we substitute the expressions for base and area into the triangle's area formula from our exercise, we derive a quadratic equation in terms of height 'h'. Solving these equations often requires methods such as factoring, using the quadratic formula, or completing the square. Regarding the exercise in question, the quadratic equation derived is \(2h^2 - 16h - 48 = 0\), which, upon solving, would yield the height of the triangle. Mastery of solving quadratic equations is crucial in cracking such mathematical challenges.

Mathematical Problem-Solving

Mathematical problem-solving is a critical skill that involves understanding the problem, devising a plan, carrying out that plan, and checking the results. It's a step-by-step process that requires analytical thinking and the application of mathematical concepts.

In our triangle area problem, problem-solving starts with interpreting the given information and expressing it using algebraic expressions. After establishing the relationship between base and height, the next step is to incorporate these into the triangle area equation, which then leads to a quadratic equation. Solving this requires algebraic manipulation and the use of problem-solving strategies to find the height 'h', illustrating how mathematical problem-solving encompasses understanding the properties of shapes (triangle geometry), algebraic manipulation (algebraic expressions and equations), and methods for solving equations (solving quadratic equations).