Problem 42 - Fundamentals of Mathematics

Question

A 5 -foot woman casts a 9 -foot shadow at a particular time of the day. How tall is a person that casts a 10.8 -foot shadow at the same time of the day?

Step-by-Step Solution

Answer

The person is 6 feet tall.

Step 1: Identify the Known Ratios

We know that a 5-foot woman casts a 9-foot shadow. Therefore, the ratio of her height to her shadow length is \( \frac{5}{9} \).

Step 2: Set Up the Proportion for the Unknown Height

We need to find the height of a person who casts a 10.8-foot shadow. Let this height be \( x \) feet. Set up the proportion: \( \frac{5}{9} = \frac{x}{10.8} \).

Step 3: Solve for the Unknown Height

By cross-multiplying the proportion \( \frac{5}{9} = \frac{x}{10.8} \), we have:\[ 5 \times 10.8 = 9 \times x \]Calculate \( 5 \times 10.8 = 54 \). Then divide both sides by 9:\[ x = \frac{54}{9} = 6 \].

Step 4: Conclusion

The height of the person who casts a 10.8-foot shadow is 6 feet.

Key Concepts

Understanding Ratios

In mathematics, a ratio is a way to describe the relationship or comparison between two quantities. It tells us how much of one thing there is compared to another. When dealing with proportions, identifying the ratio is the first crucial element. In our example, we have a 5-foot woman casting a 9-foot shadow. This establishes a ratio of height to shadow length as \( \frac{5}{9} \).

• The numerator represents the woman's height (5 feet).
• The denominator stands for the length of her shadow (9 feet).

When setting proportions, matching ratios helps us find missing values. Think of ratios as making two different scenarios harmonious in terms of relation.

The Role of Shadow Length

Shadow length can be affected by various factors, including the position of the sun. However, when all conditions remain constant, the relationship between an object's height and its shadow length remains proportional. In this problem, shadow length becomes pivotal because we're comparing two objects casting shadows at the same time of day. For the 5-foot woman, it is a 9-foot shadow. For the unknown height, we know the shadow is 10.8 feet long. Here, shadow length acts as a vital link that helps us compute the corresponding height using proportions. As you notice changes in shadow length, you can infer changes in either time, position, or object's height if other factors remain consistent.

Solving Proportions with Cross Multiplication

Cross multiplication is a reliable method for solving proportions, making it a handy tool in proportion problems. With cross multiplication, you multiply crosswise, enabling you to solve for unknowns efficiently. Let's take the proportion from our example: \( \frac{5}{9} = \frac{x}{10.8} \).Here's how you apply cross multiplication:

• Multiply the numerator of the first ratio by the denominator of the second: \( 5 \times 10.8 = 54 \).
• Multiply the denominator of the first ratio by the numerator of the second: \( 9 \times x = 9x \).

After these steps, you equate them: \( 54 = 9x \).To solve for \( x \), divide both sides by 9, derived from simplicity in arithmetic: \( x = \frac{54}{9} = 6 \). This calculation unveils the person's height to be 6 feet, showing cross multiplication as a straightforward and effective approach for solving proportion equations.