Question

the percentage of adult height attained by a girl who is x years old can be modeled by
f(x)=61 + 34 log(x - 3)
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.
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 rapidly at a young age, and continues to increase even faster as one gets older.
○ b. height increases rapidly at a young age, stops increasing at a certain age, and then starts decreasing.
○ c. height increases at a steady rate, regardless of ones age.
○ d. height increases rapidly at a young age, and then increases more slowly.

Explanation:

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.