Problem 83 - Elementary Algebra

Question

The base of a triangle is 4 in. longer than twice the height. a. If \(h=\) height, write a polynomial expression in \(h\) that represents the base, and draw a diagram of the triangle. Do not include the units. b. Write a polynomial expression in \(h\) that represents the area.

Step-by-Step Solution

Answer

a. Base: \(2h + 4\). b. Area: \(h^2 + 2h\).

Step 1: Define Variables

Let the height of the triangle be denoted by the variable \(h\). Since the base is 4 inches longer than twice the height, express the base in terms of \(h\).

Step 2: Formulate the Base Expression

Given that the base is 4 inches longer than twice the height, the base can be written as \(2h + 4\).

Step 3: Write the Expression for the Area

The area of a triangle is given by \( \frac{1}{2} \times \text{base} \times \text{height} \). Substitute the base expression \(2h + 4\) and the height \(h\) into the area formula: \[ A = \frac{1}{2} \times (2h + 4) \times h \] Simplify the expression to: \[ A = \frac{1}{2} \times (2h^2 + 4h) \] \[ A = h^2 + 2h \]

Key Concepts

Polynomial Expressions

Polynomial expressions are a combination of variables, coefficients, and exponents. They can include constants, and typically involve sums and differences of multiple terms. Each term in a polynomial can involve a variable raised to an exponent. For example, in the polynomial expression given in the problem, we have terms involving the variable 'h'. The term \( 2h + 4 \) represents the base of the triangle in terms of the height 'h'. The height 'h' is multiplied by 2 and increased by 4 to form the base. We can also see the expression \( h^2 + 2h \) representing the area of the triangle. Here, 'h' is squared and then 2 times 'h' is added to it. Understanding how to form and manipulate polynomial expressions is essential in solving many algebra problems.

Triangle Area Formula

The area of a triangle can be found using a simple formula involving its base and height. The formula is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] In our problem, to find the area, we substitute the expression for the base and the value for the height. Remember, the base is \(2h + 4\) and the height is \(h\). Substitute into the formula: \[ \text{Area} = \frac{1}{2} \times (2h + 4) \times h \] Next, use the distributive property to expand the expression inside the parentheses: \[ \text{Area} = \frac{1}{2} \times (2h^2 + 4h) \] Then, simplify further: \[ \text{Area} = h^2 + 2h \] This final polynomial \(h^2 + 2h\) represents the area of the triangle in terms of its height 'h'.

Defining Variables in Algebra

Defining variables is a fundamental skill in algebra. It helps translate real-world situations into mathematical expressions. In the given problem, we start by defining 'h' as the height of the triangle. Then, we describe the base in relation to this height by noting that the base is 4 inches longer than twice the height. Mathematically, this is represented as \(2h + 4\). Here are some tips for defining variables:

• Identify the quantities that change.
• Choose a letter or symbol to represent each quantity.
• Be consistent with your variables throughout the problem.
• Write expressions that relate the different variables to each other.

By clearly defining variables, you can set up equations and expressions that accurately describe the problem, making it easier to find a solution.