Problem 13 - Intermediate Algebra

Question

Statue of Liberty. From the foundation of the large base on which it sits to the top of the torch, the Statue of Liberty National Monument measures 305 feet. The base is 3 feet taller than the statue. Find the height of the base and the height of the statue. (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The statue is 151 feet tall, and the base is 154 feet tall.

Step 1: Define Variables

Let's define the height of the statue as \( S \) feet. The height of the base, being 3 feet taller than the statue, can be represented as \( S + 3 \) feet.

Step 2: Set Up Equation

The total height from the foundation to the top of the torch is the sum of the statue's height and the base's height. Thus, we can write the equation as \( S + (S + 3) = 305 \).

Step 3: Simplify the Equation

Combine the terms in the equation: \( 2S + 3 = 305 \).

Step 4: Solve for S

To solve for \( S \), subtract 3 from both sides of the equation, getting \( 2S = 302 \). Then divide both sides by 2, resulting in \( S = 151 \).

Step 5: Calculate the Base's Height

Since the base is 3 feet taller than the statue, add 3 to \( S \). Therefore, the height of the base is \( 151 + 3 = 154 \) feet.

Key Concepts

Linear Equations

Linear equations are mathematical statements that describe a straight line when graphed on a coordinate plane. These equations have the general form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. In the context of this exercise, the total height of the Statue of Liberty is expressed as a linear equation: \( S + (S + 3) = 305 \). Here, \( S \) and \( S + 3 \) denote the statue and its base, respectively.
Linear equations are used to find unknown values in problems by setting up an equation that models the relationship between different quantities.
These questions often involve balancing operations, such as addition or subtraction on both sides, to isolate the variable and solve the equation. This makes linear equations a fundamental tool in algebra for solving general word problems.

Variables Definition

Defining variables is the first crucial step in solving algebra word problems. Variables are symbols that represent unknown values or quantities. In this exercise, we defined \( S \) to symbolize the stature's height. By using variables:

• We can convert words into math.
• It allows expressing relationships between different quantities.
• Ensures clarity and precision when we set up equations.

This step of defining variables makes subsequent steps, like setting up and solving equations, much more manageable. This abstraction helps to simplify complex problems into something more straightforward.

Equation Solving

Equation solving is the process of finding the value of the variable that makes the equation true. Once we have our linear equation from the previous sections, we employ the strategy of balancing to solve it. Here's a breakdown of how that works in our problem:

• Step 1: Start from the equation \( 2S + 3 = 305 \).
• Step 2: Subtract 3 from both sides to get \( 2S = 302 \).
• Step 3: Divide both sides by 2 to isolate \( S \), resulting in \( S = 151 \).

This step-by-step approach of breaking down operations helps solve various algebra problems, reinforcing the idea that the value found should satisfy the original equation once plugged back in. Equation solving thus becomes iterative, moving us closer to the solution through systematic simplification.

Mathematical Modeling

Mathematical modeling involves depicting a real-world situation using mathematical concepts and formulae. It is at the heart of solving word problems in algebra. In this particular exercise, we modeled the physical structure of the Statue of Liberty using an algebraic equation:

• We began by understanding the problem scenario.
• Next, defined the height relationship between the statue and the base.
• Then formulated an equation \( S + (S + 3) = 305 \) based on these relationships.

Mathematical modeling converts a statement about physical heights into something solvable with math. This facilitates understanding and solving for unknown variables, making it indispensable in both academic settings and real-world applications, where predicting outcomes based on known data is essential.