the percentage of adult height attained by a girl who is x years old can be modeled by\nf(x)=61 + 34 log(x…

the percentage of adult height attained by a girl who is x years old can be modeled by\nf(x)=61 + 34 log(x - 3)\nwhere 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.\nb. why was a logarithmic function used to model the percentage of adult height attained by a girl from ages 5 to 15, inclusive?\n○ a. height increases rapidly at a young age, and continues to increase even faster as one gets older.\n○ b. height increases rapidly at a young age, stops increasing at a certain age, and then starts decreasing.\n○ c. height increases at a steady rate, regardless of ones age.\n○ d. height increases rapidly at a young age, and then increases more slowly.

the percentage of adult height attained by a girl who is x years old can be modeled by\nf(x)=61 + 34 log(x - 3)\nwhere 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.\nb. why was a logarithmic function used to model the percentage of adult height attained by a girl from ages 5 to 15, inclusive?\n○ a. height increases rapidly at a young age, and continues to increase even faster as one gets older.\n○ b. height increases rapidly at a young age, stops increasing at a certain age, and then starts decreasing.\n○ c. height increases at a steady rate, regardless of ones age.\n○ d. height increases rapidly at a young age, and then increases more slowly.

Answer

Brief Explanations:

Logarithmic functions have a characteristic shape where the rate of increase slows down over time. In the context of a girl's height from ages 5 - 15, young children (around age 5) grow relatively rapidly. As they get older (approaching age 15), the rate of height increase slows. This matches the behavior of a logarithmic function (y = a + b\log(x - c)) (where (a = 61), (b=34), (c = 3) in the given function (f(x)=61 + 34\log(x - 3))).

  • Option A: Logarithmic functions do not show an accelerating growth rate. The derivative of (y=\log(u)) (using the chain - rule (y^\prime=\frac{b}{(x - c)\ln(10)}) for (y=a + b\log(x - c))) shows a decreasing rate of change.
  • Option B: Logarithmic functions are increasing functions (for (x>c) in (y = a + b\log(x - c)) when (b>0)). They do not start decreasing.
  • Option C: A steady - rate increase would be modeled by a linear function (y=mx + n), not a logarithmic function. The rate of change of a logarithmic function (y = a + b\log(x - c)) is (\frac{b}{(x - c)\ln(10)}), which is not constant.
  • Option D: This matches the behavior of a logarithmic function. The function (f(x)=61 + 34\log(x - 3)) has a positive coefficient ((b = 34>0)) for the logarithmic term. The derivative (f^\prime(x)=\frac{34}{(x - 3)\ln(10)}) is positive (so the function is increasing) and decreases as (x) increases (since as (x) gets larger, ((x - 3)) gets larger and (\frac{34}{(x - 3)\ln(10)}) gets smaller).

Answer:

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