Problem 8 - Algebra A Combined Function

Question

Substitute the given values into each given formula and solve for the unknown variable. $$ V=\frac{1}{3} A h ; \quad V=45, h=5 $$

Step-by-Step Solution

Answer

The area of the base \( A \) is 27.

Step 1: Identify the Given Values

We are given the formula for the volume of a triangular prism, \( V = \frac{1}{3} A h \), where \( V = 45 \) and \( h = 5 \). Our goal is to solve for \( A \), the area of the base of the prism.

Step 2: Substitute Known Values into the Formula

Substitute the given values into the formula: \( 45 = \frac{1}{3} A \times 5 \).

Step 3: Simplify the Equation

Simplify the equation by multiplying the constants: \( 45 = \frac{1}{3} \times 5 \times A \) becomes \( 45 = \frac{5}{3} A \).

Step 4: Solve for the Unknown Variable

To isolate \( A \), multiply both sides of the equation by the reciprocal of \( \frac{5}{3} \), which is \( \frac{3}{5} \). This gives: \( A = 45 \times \frac{3}{5} = 27 \).

Key Concepts

Formula Substitution

When tackling algebra problems, formula substitution is a key method to find unknowns. In this context, we use a known formula and plug in given values to simplify and solve the problem.

For example, we have the formula for the volume of a triangular prism: \( V = \frac{1}{3} A h \). Here, \( V \) is the volume, \( A \) is the area of the base, and \( h \) is the height of the prism. Substitute the values \( V = 45 \) and \( h = 5 \) into the formula, replacing the respective variables with the known numbers.

This step allows us to change our general formula into a specific equation: \( 45 = \frac{1}{3} A \times 5 \). By substituting these values, we are set up to solve for the unknown, \( A \).

Using formula substitution involves:

• Identifying the given values and the formula.
• Replacing the variables with these values.

This process is essential to systematically approach and simplify algebra problems.

Solving for Unknowns

Solving for unknowns is a fundamental process in algebra. After substituting the known values into a formula, the next step is to isolate the unknown variable.

In our example, after substitution, the equation becomes \( 45 = \frac{5}{3} A \). Here, \( A \) is our unknown. To solve for \( A \), we need to manipulate the equation to get \( A \) by itself on one side. This can be achieved through a series of algebraic operations.

The equation \( 45 = \frac{5}{3} A \) can be rewritten by multiplying both sides by the reciprocal of \( \frac{5}{3} \), which is \( \frac{3}{5} \). This clears the fraction and isolates \( A \):

• Multiply both sides by \( \frac{3}{5} \).
• The equation becomes: \( A = 45 \times \frac{3}{5} \).
• After calculation: \( A = 27 \).

Isolating the variable like this is crucial in solving for unknowns efficiently.

Triangular Prism Volume

The concept of calculating the volume of a triangular prism is rooted in geometry. It requires finding the space inside this three-dimensional shape.

The volume formula is \( V = \frac{1}{3} A h \), where \( A \) is the area of the triangular base, \( h \) is the height, and \( V \) is the volume. This formula reflects the idea that the volume is a measure of how much space the shape occupies.

To apply this formula, imagine filling the prism with a substance. The volume's mathematical representation uses the base area, scaled by the height, then adjusted by dividing by 3 for the prism shape.Finding the volume of a triangular prism involves:

• Calculating the base area (\( A \)).
• Using the height (\( h \)) to extend this area upwards.
• Incorporating the \( \frac{1}{3} \) to adjust for the prism's shape.

Understanding this formula allows one to compute the volume of any triangular prism by using specific base area and height values.