Questions: A child is 20 inches long at birth. If we consider that the baby will grow at a rate proportional to its body size until adulthood, then the percentage of her adult height attained can be modeled by the following logarithmic function. f(x)=20+47 log (x+2) where x represents her age in years and f(x) represents the percentage of her adult height reached at age x. At what age will the child reach 95% of her adult height? Round your answer to the nearest whole year, if necessary.

Solution Steps

The given function is \( f(x) = 20 + 47 \ln(x + 2) \), where \( x \) represents the child's age in years and \( f(x) \) represents the percentage of their adult height reached at age \( x \).

We need to find the age \( x \) when the child reaches 90% of their adult height. Therefore, we set \( f(x) = 90 \): \[ 90 = 20 + 47 \ln(x + 2) \]

Subtract 20 from both sides to isolate the logarithmic term: \[ 90 - 20 = 47 \ln(x + 2) \] \[ 70 = 47 \ln(x + 2) \]

Divide both sides by 47 to solve for the logarithm: \[ \frac{70}{47} = \ln(x + 2) \] \[ \ln(x + 2) = \frac{70}{47} \]

Exponentiate both sides to solve for \( x + 2 \): \[ x + 2 = e^{\frac{70}{47}} \]

Subtract 2 from both sides to solve for \( x \): \[ x = e^{\frac{70}{47}} - 2 \]

Use a calculator to find the numerical value of \( x \): \[ x \approx e^{1.489} - 2 \] \[ x \approx 4.43 - 2 \] \[ x \approx 2.43 \]

Final Answer