Questions: The percentage of adult height attained by a girl who is x years old can be modeled by f(x)=62+35 log (x-5) where x represents the girl's age (from 5 to 15 ) and f(x) represents the percentage of her adult height. Complete parts (a) and (b) below. a. According to the model, what percentage of her adult height has a girl attained at age 12 ? A girl has attained % of her adult height by age 12 . (Do not round until the final answer. Then round to the nearest tenth as needed.) b. Why was a logarithmic function used to model the percentage of adult height attained by a girl from ages 5 to 15, inclusive? A. Height increases at a steady rate, regardless of one's age. B. Height increases rapidly at a young age, stops increasing at a certain age, and then starts decreasing. C. Height increases rapidly at a young age, and then increases more slowly. D. Height increases rapidly at a young age, and continues to increase even faster as one gets older.

The percentage of adult height attained by a girl who is x years old can be modeled by

f(x)=62+35 log (x-5)

where x represents the girl's age (from 5 to 15 ) and f(x) represents the percentage of her adult height. Complete parts (a) and (b) below.
a. According to the model, what percentage of her adult height has a girl attained at age 12 ?

A girl has attained % of her adult height by age 12 .
(Do not round until the final answer. Then round to the nearest tenth as needed.)
b. Why was a logarithmic function used to model the percentage of adult height attained by a girl from ages 5 to 15, inclusive?
A. Height increases at a steady rate, regardless of one's age.
B. Height increases rapidly at a young age, stops increasing at a certain age, and then starts decreasing.
C. Height increases rapidly at a young age, and then increases more slowly.
D. Height increases rapidly at a young age, and continues to increase even faster as one gets older.

Transcript text: The percentage of adult height attained by a girl who is $x$ years old can be modeled by \[ f(x)=62+35 \log (x-5) \] where $x$ represents the girl's age (from 5 to 15 ) and $f(x)$ represents the percentage of her adult height. Complete parts (a) and (b) below. a. According to the model, what percentage of her adult height has a girl attained at age $12 ?$ A girl has attained $\square$ $\%$ of her adult height by age 12 . (Do not round until the final answer. Then round to the nearest tenth as needed.) b. Why was a logarithmic function used to model the percentage of adult height attained by a girl from ages 5 to 15, inclusive? A. Height increases at a steady rate, regardless of one's age. B. Height increases rapidly at a young age, stops increasing at a certain age, and then starts decreasing. C. Height increases rapidly at a young age, and then increases more slowly. D. Heiaht increases rapidly at a vouna aqe, and continues to increase even faster as one qets older.

Solution

Solution Steps

To solve part (a), we need to evaluate the function $ f(x) = 62 + 35 \log(x-5) $ at $ x = 12 $. This will give us the percentage of adult height attained by a girl at age 12. For part (b), we need to understand the behavior of logarithmic functions: they increase rapidly at first and then the rate of increase slows down, which matches the typical growth pattern of height in children.

Step 1: Evaluate the Function

To find the percentage of adult height attained by a girl at age $ x = 12 $, we substitute $ x $ into the function $ f(x) = 62 + 35 \log(x - 5) $:

\[ f(12) = 62 + 35 \log(12 - 5) = 62 + 35 \log(7) \]

Step 2: Calculate the Logarithm

Next, we calculate $ \log(7) $:

\[ \log(7) \approx 0.8451 \]

Step 3: Complete the Calculation

Now, we substitute $ \log(7) $ back into the function:

\[ f(12) = 62 + 35 \times 0.8451 \approx 62 + 29.5785 \approx 91.5785 \]

Rounding to the nearest tenth, we find:

\[ f(12) \approx 91.6 \]

Step 4: Explain the Model Choice

The logarithmic function is used to model the percentage of adult height attained because height increases rapidly at a young age and then increases more slowly as the girl ages. This behavior aligns with option C.