Problem 20 - College Algebra with Modeling and Visualization

Question

Find the area of the triangle with base \(b\) and height \(h .\) \(b=y+1, h=2 y\)

Step-by-Step Solution

Answer

The area of the triangle is \( y^2 + y \).

Step 1: Formula Identification

Recall the formula for the area of a triangle. The area \( A \) of a triangle can be calculated using the formula: \( A = \frac{1}{2} \times \, \text{base} \times \, \text{height} \).

Step 2: Substitute Values

Substitute the given expressions for the base \( b \) and height \( h \) into the formula. Given \( b = y + 1 \) and \( h = 2y \), substitute these into the area formula: \( A = \frac{1}{2} \times (y+1) \times (2y) \).

Step 3: Simplify the Expression

Simplify the expression for the area. \( A = \frac{1}{2} \times (y+1) \times 2y = \frac{1}{2} \times (2y^2 + 2y) \).

Step 4: Final Simplification

Distribute and simplify the expression: \( A = y^2 + y \).

Step 5: Result

Conclude that the area of the triangle is expressed in terms of \( y \) as \( A = y^2 + y \).

Key Concepts

Geometry

Understanding the area of a triangle is a fundamental concept in geometry. A triangle is a three-sided polygon, and calculating its area helps us understand the space it occupies. The formula to calculate the area of a triangle is given by:

• \( A = \frac{1}{2} \times \text{base} \times \text{height} \)

This formula derives from taking half of the product of the base and height. The base of a triangle can be any one of its sides, but it is typically easiest to choose the side parallel to the horizontal line for calculation purposes. The height is the perpendicular distance from this base to the opposite vertex.
In our exercise, the triangle is defined algebraically with a base denoted as \( b = y + 1 \) and a height \( h = 2y \). This allows for flexibility, as the values of the base and height can change depending on the value of \( y \). Understanding how these dimensions affect the area helps in solving more complex geometry problems.

Algebraic Expressions

Algebraic expressions allow us to generalize mathematical relationships with variables. In the triangle area problem, the base and height are expressed as algebraic expressions, making the calculation dynamic.

• The base is \( b = y + 1 \)
• The height is \( h = 2y \)

These expressions represent the lengths of the triangle's base and height depending on the value of \( y \). By substituting these into the area formula, we have an algebraic expression for the area:

• \( A = \frac{1}{2} \times (y+1) \times (2y) \)

Simplifying these expressions is key. When multiplying algebraic expressions, remember to use the distributive property, which allows you to expand expressions:

• \( A = \frac{1}{2} \times (2y^2 + 2y) \)

This eventually simplifies to \( A = y^2 + y \), an elegant representation of the triangle's area depending on \( y \). This expression reflects how varying \( y \) alters the size of the triangle.

Problem-Solving Steps

Breaking down problems into manageable steps is crucial for successful problem-solving in mathematics. Each step builds upon the previous one, structuring a logical path to the solution. Here's a brief overview of how we can solve for the area of a triangle using algebra:

• Step 1: Identify the Formula. Recognize which formula is relevant: for this exercise, it's the area of a triangle formula \( A = \frac{1}{2} \times \text{base} \times \text{height} \).
• Step 2: Substitute Values. Replace the variables in the formula with given expressions: here, \( b = y + 1 \) and \( h = 2y \).
• Step 3: Simplify the Expression. Use algebraic rules to simplify the expression for the area: distribute and combine like terms.
• Step 4: Final Simplification. Ensure the expression is as simple as possible, resulting in \( A = y^2 + y \).

Following these steps not only helps in this specific problem but also develops a structured approach to tackle similar algebraic and geometric problems. Consistent practice with this method enhances problem-solving skills across various mathematical contexts.