Problem 1 - College Algebra with Modeling and Visualization

Question

Evaluate the function for the indicated inputs and interpret the result. \(A(5,8),\) where \(A(b, h)=\frac{1}{2} b h(A\) computes the area of a triangle with base \(b\) and height \(h\) )

Step-by-Step Solution

Answer

The area of the triangle is 20 square units.

Step 1: Identify the Function and Inputs

The function provided is for the area of a triangle, given by \(A(b, h) = \frac{1}{2} b h\). The inputs indicated are \(b = 5\) and \(h = 8\).

Step 2: Substitute the Inputs into the Function

Substitute the base \(b = 5\) and height \(h = 8\) into the function: \(A(5, 8) = \frac{1}{2} \times 5 \times 8\).

Step 3: Perform the Calculation

Calculate the product: \(\frac{1}{2} \times 5 \times 8 = \frac{1}{2} \times 40 = 20\).

Step 4: Interpret the Result

The result of the calculation is \(20\). This means that the area of the triangle, with the given base and height, is 20 square units.

Key Concepts

Functions

In algebra, functions are special types of equations that establish a relationship between two sets of variables. These variables are known as inputs and outputs. To simplify this, a function can be thought of as a machine that takes an input, performs a calculation, and then spits out a result or output.
For our specific function, we are looking at one that calculates the area of a triangle. It is written as \( A(b, h) = \frac{1}{2} b h \), where \( b \) represents the base of the triangle and \( h \) represents the height.

• The function tells you exactly what to do with the input numbers. Here, it instructs you to multiply the base by the height, then take half of that product.
• The result is the area of the triangle, which is the number of square units inside it.

Understanding functions and how to use them is critical for solving many types of mathematical problems.

Area of a Triangle

The area of a triangle is a measure of the space inside the triangle's boundaries. It's a fundamental concept in geometry and essential for various real-world applications, such as architecture and engineering.
The formula to compute the area of a triangle is \( A = \frac{1}{2} b h \), where \( b \) is the base and \( h \) is the height. This formula originates from the fact that a triangle is essentially half of a rectangle with the same base and height. Here's how it works:

• Find the base, \( b \), of the triangle, which is a side of the triangle you'll use as the reference line.
• Measure the height, \( h \), which is the perpendicular distance from the base to the opposite vertex.
• Apply the formula by multiplying the base and height, and then halving the result to find the area.

This straightforward formula helps determine how much space a triangle occupies on a flat surface.

Substitution Method

The substitution method is a straightforward technique in algebra that involves replacing variables with their corresponding values, turning an expression into one that is easier to solve. This approach is widely used whenever you know specific values for your variables.
In our example with the function \( A(b, h) = \frac{1}{2} b h \), we used substitution to find the area of a triangle with specific measurements. Here’s how:

• Identify the function you are dealing with and understand the values you need to substitute. In our case, these were the base \( b = 5 \) and the height \( h = 8 \).
• Substitute these known values for the respective variables in the function. Thus, \( A(5, 8) = \frac{1}{2} \times 5 \times 8 \).
• Perform the calculations to get the final result, which was 20 square units for the area in this case.

This method makes complex problems manageable by breaking them down into simpler arithmetic operations.