Problem 58 - Elementary Algebra

Question

The area of a triangle is 48 square inches. If the base is 2 times the height, then find the length of the base.

Step-by-Step Solution

Answer

The base length is \( 8\sqrt{3} \) inches.

Step 1: Understanding the Formula for Area

The area of a triangle is given by the formula \( A = \frac{1}{2} \times \text{base} \times \text{height} \). For this problem, we know that the area \( A = 48 \) square inches.

Step 2: Relating Base and Height

According to the problem, the base \( b \) is 2 times the height \( h \). We can write this as the equation \( b = 2h \).

Step 3: Setting Up the Area Equation

Substitute the relationship between base and height into the area formula: \( 48 = \frac{1}{2} \times (2h) \times h \). Simplifying, we get \( 48 = h^2 \).

Step 4: Solving for Height

To find \( h \), take the square root of both sides of the equation \( 48 = h^2 \). We get \( h = \sqrt{48} \). Simplifying, \( h = 4\sqrt{3} \).

Step 5: Substituting Back to Find Base

Use the relationship \( b = 2h \). Substitute \( h = 4\sqrt{3} \) into this equation: \( b = 2 \times 4\sqrt{3} \). Simplifying gives \( b = 8\sqrt{3} \).

Key Concepts

Understanding the Triangle Formula

To find the area of a triangle, we use a simple yet powerful formula. The formula is expressed as \( A = \frac{1}{2} \times \text{base} \times \text{height} \). This helps in calculating how much surface the triangle covers. The reason for this formula is that a triangle is essentially half of a rectangle, which you will notice when you visualize it folding over the base. This understanding enables us to link the base and height smoothly in mathematical expressions.

In our problem, the area is given as 48 square inches, providing a concrete value that we can work with to solve for unknowns.

Understanding Base and Height Relationship

In many geometry problems, the base and height of a triangle are related in some way. Here, we're told that the base \( b \) is 2 times the height \( h \). This relationship can be written as the equation \( b = 2h \).

This equation signifies the dependency between base and height. Understanding such relationships lets us replace one variable with another in equations, simplifying the problem-solving process.

• If the base were 3 times the height, the equation would be \( b = 3h \).
• If the height were half of the base, it would be \( h = \frac{1}{2}b \).

Such relationships are quite helpful in real-world applications, like construction and design.

Steps to Solving Algebraic Equations

Algebraic equations may seem daunting, but breaking them down step by step makes them manageable. First, we substitute known relationships into equations. In our example, substituting \( b = 2h \) into the area formula gives us \( 48 = \frac{1}{2} \times (2h) \times h \).

Simplifying further, we end up with \( 48 = h^2 \). Now, we solve for \( h \) by applying algebraic operations, ensuring that calculations are both logical and sequential:

• Isolate the variable: Ensure that the variable is alone on one side of the equation.
• Perform inverse operations: Use subtraction, addition, multiplication, or division to simplify.
• Verify: Substitute your solution back into the original context to ensure accuracy.

Such a process is key in mastering algebra.

Using the Substitution Method

The substitution method is a strategic technique in mathematics used to solve problems where variables are interconnected. In the given scenario, after solving for the height \( h \), which is \( 4\sqrt{3} \), we substitute back to find the base.

Using \( b = 2h \), we replace \( h \) with \( 4\sqrt{3} \), resulting in \( b = 2 \times 4\sqrt{3} \). Simplifying this gives \( b = 8\sqrt{3} \).

This step reinstates the dependency between base and height and shows how substitution connects different parts of a problem, allowing for a complete, accurate solution.

• It helps when multiple equations share variables, creating a seamless solution process.
• Can be applied in different fields, including physics and engineering, where systems have interconnected variables.

Utilizing substitution enhances problem-solving efficiency and clarity.