Question

the percentage of adult height attained by a girl who is x years old can be modeled by f(x)=61 + 36 log(x - 4) where x represents the girls 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 93.5 % 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 continues to increase even faster as one gets older.
d. height increases rapidly at a young age, and then increases more slowly.

Explanation:

Step1: Substitute x = 12 into the function

Substitute \(x = 12\) into \(f(x)=61 + 36\log(x - 4)\), we get \(f(12)=61+36\log(12 - 4)=61 + 36\log(8)\).

Step2: Calculate the value of the logarithm

We know that \(\log(8)=\log(2^{3}) = 3\log(2)\approx3\times0.3010 = 0.9030\). Then \(36\log(8)=36\times0.9030 = 32.508\).

Step3: Calculate f(12)

\(f(12)=61+32.508=93.508\approx93.5\)

For part (b):

Logarithmic functions are often used to model situations where a quantity increases rapidly at first and then the rate of increase slows down. In the context of a girl's height - growth from ages 5 to 15, height increases rapidly at a young age and then increases more slowly as she gets older.

Answer:

a. 93.5
b. D. Height increases rapidly at a young age, and then increases more slowly.