Problem 114
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. 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
Verified Answer
The girl at age ten has attained approximately 105.1% of her adult height.
1Step 1: Identify the Relevant Variables
In our case, the age of the girl \(x\) is the independent variable and the percentage of height gained \(f(x)\), is the dependent variable. We are given \(x = 10\) years.
2Step 2: Setup the Given Function
Express the given function \(f(x) = 62 + 35\ \log(x - 4)\) and substitute \(x = 10\) into the function. Therefore, the expression becomes \(f(10) = 62 + 35\ \log(10 - 4)\).
3Step 3: Simplify the Logarithm
Simplify the logarithm in the function. \(10 - 4 = 6\), thus the function is now \(f(10) = 62 + 35\ \log(6)\).
4Step 4: Solve the Logarithm
Calculate the logarithm, \(35 \log(6)\), using an appropriate mathematical tool and then add 62 to the calculated logarithm.
5Step 5: Rounding the Result
Round off the answer to the nearest tenth as instructed.
Key Concepts
Independent VariableDependent VariableSolving Logarithms
Independent Variable
In mathematics, the independent variable is a significant concept as it is the variable that is controlled or changed to observe its effects on another variable, known as the dependent variable. In our exercise, the independent variable is the age of the girl, denoted by \(x\). This means that as age changes, it is expected to have an effect on another aspect associated with it, in this case, the percentage of her adult height achieved.
When considering a real-world scenario such as the growth of a child, the independent variable, age, does not depend on anything else within this specific context. It is clear and objective, measured as each year passes.
Understanding which variable is independent helps us focus on what can be changed and what its likely impact may be on the corresponding dependent variables. This distinction is critical when setting up models that seek to predict outcomes based on changing conditions.
When considering a real-world scenario such as the growth of a child, the independent variable, age, does not depend on anything else within this specific context. It is clear and objective, measured as each year passes.
Understanding which variable is independent helps us focus on what can be changed and what its likely impact may be on the corresponding dependent variables. This distinction is critical when setting up models that seek to predict outcomes based on changing conditions.
Dependent Variable
The dependent variable responds to the independent variable and is what is measured within an experiment or situation. In this context, it represents the percentage of adult height a girl has reached by her current age, indicated by \(f(x)\). It’s important to grasp that the dependent variable is dependent on or influenced by changes in the independent variable, which here is age \(x\).
The percentage attained of adult height is a dynamic number that reflects growth over time, or as age increases. This real-time growth measurement helps provide valuable insights into whether a growth pattern matches the expected biological norms.
Having a firm understanding of the dependent variable is essential for comprehending how changes in the independent variable, such as age increments, affect outcomes such as physical development, as captured by this percentage of adult height function.
The percentage attained of adult height is a dynamic number that reflects growth over time, or as age increases. This real-time growth measurement helps provide valuable insights into whether a growth pattern matches the expected biological norms.
Having a firm understanding of the dependent variable is essential for comprehending how changes in the independent variable, such as age increments, affect outcomes such as physical development, as captured by this percentage of adult height function.
Solving Logarithms
Logarithms are mathematical operations that help solve equations involving exponential growth or decay by transforming multiplicative processes into additive ones. In the exercise provided, the function includes a logarithmic expression, which is \( \log(x - 4) \). This part of the equation emphasizes the rate of growth concerning the girl’s age.
To solve logarithms practically, we need to simplify the equation, as done by first calculating \(10 - 4\) to get 6. Therefore, the expression transforms to \(35 \log(6)\). Logarithms require a calculator or computational tool to find an approximate numerical result.
Once the logarithm is calculated, continue by adding the constant 62 to the result of \(35 \times \log(6)\) to determine the overall percentage of adult height. Rounding the final result to the nearest tenth, as instructed, ensures precision and clarity in the physical interpretation of mathematical outcomes. Understanding logarithms and their application is essential in navigating studies associated with growth, change, and other real-world scenarios involving exponential models.
To solve logarithms practically, we need to simplify the equation, as done by first calculating \(10 - 4\) to get 6. Therefore, the expression transforms to \(35 \log(6)\). Logarithms require a calculator or computational tool to find an approximate numerical result.
Once the logarithm is calculated, continue by adding the constant 62 to the result of \(35 \times \log(6)\) to determine the overall percentage of adult height. Rounding the final result to the nearest tenth, as instructed, ensures precision and clarity in the physical interpretation of mathematical outcomes. Understanding logarithms and their application is essential in navigating studies associated with growth, change, and other real-world scenarios involving exponential models.
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