Problem 1 - Algebra A Combined Function

Question

Substitute the given values into each given formula and solve for the unknown variable. $$ A=b h ; \quad A=45, b=15 $$

Step-by-Step Solution

Answer

The height \( h \) is 3.

Step 1: Write Down the Formula

The formula given is for the area of a rectangle, which is \( A = b \cdot h \), where \( A \) is the area, \( b \) is the base, and \( h \) is the height.

Step 2: Substitute Known Values into Formula

Substitute \( A = 45 \) and \( b = 15 \) into the formula: \( 45 = 15 \cdot h \).

Step 3: Solve for the Unknown Variable

To solve for \( h \), divide both sides of the equation by 15: \( h = \frac{45}{15} \).

Step 4: Calculate the Result

Perform the division to find the value of \( h \): \( h = 3 \).

Key Concepts

Substitution in Equations

Understanding the concept of substitution in equations is essential for solving problems involving unknown variables. In mathematical equations, substitution means replacing variables with known values to simplify and solve the equation. Think of it like filling a gap with a known piece of information to make the entire statement clearer.

In our example, we had the equation for the area of a rectangle, which is represented as \( A = b \cdot h \). The goal was to find the height \( h \). To do this, we substituted the known values: area \( A = 45 \) and base \( b = 15 \), into the equation.

Substituting these values transforms the formula into a solvable equation, \( 45 = 15 \cdot h \). This way, substitution sets us up for the next step, which is solving the equation. By substituting, we laid the groundwork to figure out what \( h \) must be.

Solving Equations

Once substitution has been done, the next step is solving the equation. Solving an equation typically involves isolating the unknown variable on one side of the equation to determine its value.

For our example, after substituting the known values, the equation becomes \( 45 = 15 \cdot h \). Our goal is to isolate \( h \).

We do this by performing operations that would leave \( h \) alone on one side of the equation:

• Here, divide both sides by 15 to cancel out the base \( b \) on the right-hand side.
• This simplifies the equation to \( h = \frac{45}{15} \).

The division helps isolate \( h \), allowing us to solve it and find the exact value of the height of the rectangle.

Area Calculation

Area calculation is a fundamental concept in geometry that measures the extent of a shape's surface. In the context of rectangles, the area is determined by multiplying the base by the height, as indicated by the formula \( A = b \cdot h \). Here, the base \( b \) and height \( h \) are the rectangle's dimensions.

Calculating the area helps understand the space occupied within the boundaries of a 2D shape. In practical terms, knowing how to calculate area can be essential in everyday tasks, such as determining how much paint is needed for a wall.

In our problem, we knew the area \( A = 45 \) and the base \( b = 15 \), but needed to find the height \( h \). Solving for \( h \) involved confirming that the calculated height would indeed provide the given area when multiplied by the base. In this exercise, the division \( \frac{45}{15} = 3 \)
confirmed the height, showing how substitution, solving equations, and understanding area calculation are interconnected in geometry.