Problem 113 - College Algebra

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. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age \(13 ?\)

Step-by-Step Solution

Answer

After completing calculations, we find that a girl has attained approximately \(95.3\%\) of her adult height at age 13.

Step 1: Analyze the given function

The function provided is \(f(x)=62+35 \log (x-4)\). This function is used to estimate the percentage of adult height that a girl will have reached at age \(x\). The base of the logarithm in this case is 10, according to convention when the base is not given explicitly.

Step 2: Substitute the value \(x=13\) into the function

To find out the percentage of the adult height a girl reaches at the age of 13, the value \(13\) should be plugged into the function. Thus, \(f(13)=62+35 \log (13-4)\). Note that the calculation \(13-4\) is done first due to the order of operations.

Step 3: Calculate the logarithm and complete the calculation

Continuing from the previous step, we caluculate the logarithm and complete the calculation: \(f(13) = 62+35 \log (9)\). \(\log (9)\) is approximate \(0.95\) when rounded to two decimal places. Hence, \(f(13) = 62 + 35 * 0.95\). Carrying out the multiplication and addition, gives an approximation for \(f(13)\).

Key Concepts

Algebraic Modeling

Algebraic modeling is a method used to represent real world situations with algebraic expressions or equations. Taking our exercise for example, the growth in height of a girl as she ages is represented using the function

\[f(x)=62+35 \log (x-4)\]

Here, age is the variable that we change, and the function gives us the corresponding percentage of her adult height attained. Algebraic models like these turn empirical data and observed patterns into mathematical formats that can be analyzed and used to make predictions. Effective models require understanding the underlying principles and accurately representing relationships between quantities with mathematical symbols and operations.

Logarithmic Expressions

Logarithmic expressions are the inverse operations of exponentiation and have countless applications in real-world contexts, including the modeling of growth and decay. In the function given in our exercise,

\[\log (x-4)\]

shows up within the model. The use of the logarithm here is integral for the equation, that models the non-linear pattern of growth in height. Students need to comprehend how to work with logarithms, including understanding their properties, how to simplify them, and using a calculator to find their numerical values. In basic algebra, the logarithm base 10 is commonly used, and this is implied when no base is explicitly stated.

Percentages in Algebra

Percentages are another frequently encountered concept in algebra. They describe how one quantity relates to another in terms of hundredths. In our exercise, we are asked to find the percentage of the adult height of a girl at a certain age. The equation given translates the age (minus 4) into a percentage. When dealing with percentages in algebra, it's important to remember that we are often looking for a part of a whole, and the equation or function in use should reflect that relationship. Comprehending how percentages function within algebraic expressions allows students to solve a range of problems, from those involving growth, like in our exercise, to financial calculations among many other applications.

Mathematical Functions

Mathematical functions are paramount to representing relationships between quantities in a precise and predictable way. A function takes an input, processes it through a formula, and produces an output. In our case, the function

\[f(x)=62+35 \log (x-4)\]

is a specific type of function known as a logarithmic function, reflecting a pattern in the growth of a girl's height. Understanding functions involves recognizing the types (linear, quadratic, exponential, etc.), properties such as domain and range, and how alterations to the function's equation can affect its graph and outputs. Including the concept of a function within algebraic operations is essential, as it aids in mapping out numerous types of real-world situations quantitatively.