Problem 59 - Algebra 1: Concepts and Skills

Question

Make a sketch and write a quadratic equation to model the situation. Then solve the equation. The base of a triangle is \(x\) feet and the height is \((4+2 x)\) feet. The area of the triangle is 60 square feet. What are the dimensions of the triangle?

Step-by-Step Solution

Answer

The base of the triangle is 5 feet and the height is 14 feet.

Step 1: Write Down the Area Equation of the Triangle

Start by setting up the equation for the area of a triangle using the given variables. With \(x\) as the base and \(4 + 2x\) as the height, multiply these two values and divide by 2 to get the equation \[ (1/2)*x*(4+2x) = 60 \].

Step 2: Simplify the Equation

Next, simplify the equation to obtain a quadratic equation. This can be done by distributing the \(x\) to \((4+2x)\) and further simplifying. Doing this, one will get \[ x*(2x+4) = 60 \] or \[ 2x^2 + 4x - 60 = 0 \].

Step 3: Solve the Quadratic Equation

The equation can be factored to make it easier to solve. Divide each term by 2 to get \[ x^2 + 2x - 30 = 0 \]. Factoring this equation results in \[ (x+6)(x-5) = 0 \]. Solving this equation, \(x\) can be -6 or 5. A negative base does not make sense in physical context hence, \(x\) must be equal to 5 feet.

Step 4: Determine the Height of the Triangle

With the base of the triangle solved, determine the height by substituting \(x=5\) to the height equation \((4+2x)\). This gives us a height of 14 feet.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra, representing relationships where the highest power of the variable is two—hence the 'quad' in quadratic. They generally take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \), are constants, and \( x \) is the unknown variable.

While solving these equations can initially seem daunting, there are several methods to find the values of \( x \) that satisfy the equation. In the context of the exercise provided, the quadratic equation models a real-world situation—the area of a triangle. When addressing such problems, forming the equation is the first step that involves identifying the variables and their relationship—in this case, the base and height of a triangle related to its area.

Solving the quadratic equation might lead to one, two, or no real solutions. Importantly, consider the context to discard non-sensible solutions, like a negative measurement for the base of a triangle in real-life applications.

Factoring Quadratics

Factoring quadratics is a technique to simplify quadratic equations, making them easier to solve. The method involves rewriting the equation as a product of two binomials. For instance, the quadratic equation \( x^2 + 2x - 30 = 0 \) from the exercise can be factored into \( (x+6)(x-5) = 0 \).

The goal is to find two numbers that multiply to give the constant term, here -30, while also adding up to the linear coefficient, which is 2 in our example. Once this is done, each binomial equals zero, providing the solutions to the equation. While this method is one of the most common ways to solve a quadratic equation, it is important to note that not all quadratics are factorable using integers. In such cases, other methods like completing the square or using the quadratic formula may be used. Understanding how to factor effectively can often simplify the approach to solving many mathematical problems.

Area of a Triangle

The area of a triangle is a measure of the space contained within its three sides. The most fundamental formula for calculating the area is given by \( A = \frac{1}{2} \cdot base \cdot height \), where 'base' and 'height' are perpendicular to each other.

In the given exercise, the area relates to the quadratic equation after setting up the given dimensions of the triangle into the formula. The given values are taken as variables which are then incorporated into the equation to calculate the area. It's imperative to comprehend that the base and height of a triangle must be perpendicular. Lastly, the fact that the area of a triangle must be a positive value is significant, as it can help in determining the validity of the obtained solutions for the triangle's dimensions.