Problem 114 - Algebra and Trigonometry

Question

The percentage of adult height attained by a girl who is x years old can be modeled by $$ f(x)=62+35 \log (x-4) $$ where x represents the girl’s age (from 5 to 15) and f(x) represents the percentage of her adult height. Use the function to solve Exercises 113–114. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age ten?

Step-by-Step Solution

Answer

At the age of ten, a girl has attained approximately 84.7% of her adult height.

Step 1: Substitute Value

Substitute x = 10 in given function, f(x) = 62 + 35\log(x - 4). Computed as f(10) = 62 + 35\log(10 - 4).

Step 2: Simplify expression

Simplify f(10) = 62 + 35\log(6), compute the logarithm.

Step 3: Calculate and Round

Calculate the whole expression and round to the nearest tenth of a percent.

Key Concepts

Logarithmic Functions

Logarithmic functions are a fundamental concept in mathematics. They are often used to model growth, like in the case of the percentage of adult height attained. The logarithm represents the power to which a certain number, called the base, must be raised to obtain another number. In the example provided, the function is expressed using the common logarithm (base 10), which is denoted as \( \log \). This function can help us understand how quickly something grows in relation to its current state.

The specific equation used here is \( f(x) = 62 + 35\log(x-4) \), where \( f(x) \) represents the percentage of adult height a girl has reached by age \( x \). This model shows that growth in height doesn't happen linearly, but slows down as age increases.

When interpreting logarithmic functions in the context of real-world scenarios, remember that small changes in \( x \) when \( x \) is small can lead to larger changes in \( f(x) \). This mirrors the way human growth rapidly occurs in early years and slows as we approach maturity.

Age-related Growth

Age-related growth models describe how certain characteristics change over time, depending on age. In biological contexts, like human height growth, they are crucial because they reflect the slowing growth rate as one approaches adulthood.

The model given, \( f(x) = 62 + 35\log(x - 4) \), suggests that at very young ages, growth is not linear and tends to occur rapidly. Between ages 5 to 15, these changes slow down. This equation represents this diminishing return as age increases, showing a more substantial initial increase that flattens as \( x \) continues to grow.

Understanding this model helps predict and evaluate how much growth is expected in a certain time frame. In practical terms, when using such models, keep in mind that predicting future values always entails some level of uncertainty due to the natural variability in biological growth.

Percentage Calculations

Percentage calculations are a way to express a number as a fraction of 100, which brings useful insight in comparing or understanding data in a relatable way. They are used in this context to express how much of adult height a girl has achieved at a certain age.

In this exercise, the function \( f(x) \) outputs a percentage, such as 85.3 percent. This means the girl has attained 85.3 percent of her expected adult height by age \( x \). Working with percentages allows us to easily understand and communicate parts of a whole in everyday life, as well as in professional fields.

Calculating percentages involves simple arithmetic. For this exercise, once the value from the logarithmic function is found, it's rounded to the nearest tenth of a percent to make the information more practical and easier to understand. Remember, when handling percentages, that rounding can slightly adjust the precision of your result but generally maintains overall accuracy.