Problem 85 - Algebra and Trigonometry with Analytic Geometry

Question

A person's height will typically decrease by \(0.024\) inch each year after age 30 . (a) If a woman was 5 feet 9 inches tall at age 30 , predict her height at age 70 . (b) A 50-year-old man is 5 feet 6 inches tall. Determine an inequality for the range of heights (in inches) that this man will experience between the ages of 30 and 70 .

Step-by-Step Solution

Answer

(a) 68.04 inches; (b) Height is between 65.04 and 66 inches.

Step 1: Convert Height to Inches

First, convert the woman's height at age 30 from feet and inches to only inches. She is 5 feet 9 inches tall. Since 1 foot equals 12 inches, we calculate her height in inches as follows:\[ 5 \times 12 + 9 = 60 + 9 = 69 \text{ inches} \]

Step 2: Calculate Height Reduction Over 40 Years

Next, determine how much her height decreases between ages 30 and 70. This period spans 40 years. Each year, her height decreases by 0.024 inches:\[ 0.024 \times 40 = 0.96 \text{ inches} \]

Step 3: Predict Height at Age 70

Subtract the total height reduction from her original height at age 30 to find her height at age 70:\[ 69 - 0.96 = 68.04 \text{ inches} \]

Step 4: Convert Man's Height to Inches

For the 50-year-old man, first convert his current height of 5 feet 6 inches into inches:\[ 5 \times 12 + 6 = 60 + 6 = 66 \text{ inches} \]

Step 5: Calculate Total Height Reduction

The man experiences a height change over a total of 40 years (from age 30 to 70), with a height decrease of 0.024 inches per year:\[ 0.024 \times 40 = 0.96 \text{ inches} \]

Step 6: Determine Minimum Height

Subtract this reduction from his current height to find his predicted minimum height at age 70:\[ 66 - 0.96 = 65.04 \text{ inches} \]

Step 7: Formulate Height Inequality

Since he does not lose more than what is calculated over these years, his height is described by the inequality:\[ 65.04 \leq x \leq 66.0 \text{ inches} \] This represents the range of heights from when he was age 30 to age 70.

Key Concepts

Height Prediction

Predicting someone's height over time can help us understand age-related biological changes. Imagine a woman's height at 30 years old is 5 feet 9 inches. How will her height likely change by age 70? Predictions like these are made using simple arithmetic and linear regression concepts. Height prediction involves observing patterns, such as gradual height loss. In the given example, the woman loses 0.024 inches annually after 30. Over 40 years, that amounts to a decrease of 0.96 inches. Therefore, if she was originally 69 inches tall at 30, her predicted height would reduce to 68.04 inches by age 70. This prediction helps in understanding personal change over time, while taking aging factors into account.

Age-Related Height Change

Our bodies naturally change as we age, and this includes height. Generally, people begin to lose height after age 30. The exercise highlights a common rule: a decrease of about 0.024 inches per year. This change is attributed to factors like joint compression and posture shifts.

• The woman in the example would lose height over 40 years, totaling nearly an inch.
• A similar pattern would apply to others in different age ranges.

Understanding this can help people adjust expectations about body changes as they get older. It's important because it reflects natural, predictable processes rather than sudden changes, which can reassure those experiencing this gradual transition.

Mathematical Modeling

Mathematical modeling is a crucial tool in predicting outcomes in various fields, including human biology. Let's use our height example. By creating a formula to represent height decrease, simple models can provide significant insights.

Basics of Linear Regression

Imagine a line on a graph where the x-axis represents age and the y-axis represents height. This would likely be a downward slant to indicate height loss with aging. Linear regression helps find and predict this trend over time.

Applying the Model

For the man in our original exercise, starting at age 50 with a height of 66 inches, we can model his height loss until 70. Using a formula (\(x - (0.024 \times (70-x))\)), we predict height at different ages. Between 30 and 70, his height range becomes 65.04 to 66 inches, according to our model.Models are not only theoretical but have practical uses, offering frameworks for understanding and planning based on observed trends.