Problem 72 - Elementary Algebra

Question

At the same time of day, a father and son, standing side by side, cast a 4-foot and 2 -foot shadow, respectively. If the father is 6 feet tall, then how tall is his son?

Step-by-Step Solution

Answer

The son is 3 feet tall.

Step 1: Understanding Proportions

To solve this problem, we need to understand that the shadows cast are proportional to the heights of the objects casting them. This means that the ratio of height to shadow length for the father will be the same as the ratio of height to shadow length for the son.

Step 2: Setting Up the Equation

Let the height of the son be denoted by \( h_s \). We know the father's height is 6 feet and his shadow is 4 feet long, so his ratio is \( \frac{6}{4} \). The son's shadow is 2 feet long. The equation is: \( \frac{6}{4} = \frac{h_s}{2} \).

Step 3: Solving for the Son's Height

To find \( h_s \), we solve the equation \( \frac{6}{4} = \frac{h_s}{2} \). Cross-multiply to get \( 6 \times 2 = 4 \times h_s \), which simplifies to \( 12 = 4h_s \).

Step 4: Calculating the Son's Height

Divide both sides by 4 to isolate \( h_s \): \( h_s = \frac{12}{4} = 3 \). So, the son is 3 feet tall.

Key Concepts

Ratio

A ratio is a way to compare two quantities. It tells us how much of one thing there is compared to another. For instance, in a recipe if you use 2 cups of flour for every 1 cup of sugar, the ratio is 2:1.

In our problem, both the father and son are casting shadows, allowing us to set up a comparison between their heights and the lengths of their shadows. We use the ratio of height to shadow length. For the father, the ratio is \( \frac{6}{4} \), where 6 is his height in feet, and 4 is the shadow length. Similarly, for the son's shadow, we use the ratio \( \frac{h_s}{2} \). This setup helps us to see how proportional reasoning can give us the son's height.

Cross-Multiplication

Cross-multiplication is a method used to solve equations involving ratios or proportions. It allows us to find unknown values when we have one complete ratio and another incomplete one.

In this problem, we set up the equation \( \frac{6}{4} = \frac{h_s}{2} \). Cross-multiplication involves multiplying the numerator of one ratio by the denominator of the other, and vice versa. This means we multiply 6 by 2, and 4 by \( h_s \). As a result, we get the equation: \( 6 \times 2 = 4 \times h_s \), which simpliflies to \( 12 = 4h_s \). This step allows us to handle proportions more easily, removing denominators that we may not be able to solve directly.

Equations

Equations are mathematical statements that show the equality between two expressions. Solving equations involves finding the values of unknown variables that make the statement true.

In the shadow problem, once cross-multiplication sets up \( 12 = 4h_s \), the last step is to solve for \( h_s \). We want to isolate \( h_s \) on one side of the equation. To do this, divide both sides by 4, yielding \( h_s = \frac{12}{4} \), which gives \( h_s = 3 \). This tells us the son is 3 feet tall.

Equations help us work through problems in a structured way, often starting with setting up relationships as we did with the shadows, before transforming and simplifying them to find solutions.