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 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, stops increasing at a certain age, and then starts decreasing.
b. height increases rapidly at a young age, and continues to increase even faster as one gets older.
c. height increases rapidly at a young age, and then increases more slowly.
d. height increases at a steady rate, regardless of ones age.

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, we know that children grow rapidly in their early - to - mid - childhood years. As they approach adolescence (around the upper end of the 5 - 15 age range), the rate of height increase slows down.

• Option A is incorrect because height does not start decreasing within the 5 - 15 age range.
• Option B is incorrect because the rate of height increase does not accelerate as one gets older within the 5 - 15 age range.
• Option D is incorrect because height does not increase at a steady rate. Growth spurts and slowing of growth are common in the 5 - 15 age range.
• Option C is correct. The logarithmic function \(y = a + b\log(x - c)\) (in this case \(a = 62\), \(b = 35\), \(c = 4\)) has a graph that first rises rapidly (for smaller values of \(x\)) and then the slope of the graph (the rate of increase) gets smaller (for larger values of \(x\) in the domain \(x\in[5,15]\)), which matches the real - world observation that height increases rapidly at a young age and then more slowly.

Answer:

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