Question

researchers fit data from over 6000 fetal ultrasounds. they measured the femur length, l, (in millimeters) as a function of the fetus age, t, (in weeks) and found the formula below. calculate the rate of growth, \\( \frac { d l } { d t } \\), at \\( t = 10,15 \\), and 25 weeks. does the rate of growth of the fetus increase or decrease as it ages? \\( l = - 37.60 + 3.71 t - 6.32 \times 10 ^ { - 4 } t ^ { 3 } \\) (use integers or decimals for any numbers in the expression.) calculate the rate of growth at \\( t = 10 \\) weeks. \\( \frac { d l } { d t } = 3.52 \\) mm/week (round to two decimal places as needed.) calculate the rate of growth at \\( t = 15 \\) weeks. \\( \frac { d l } { d t } = \square \\) mm/week (round to two decimal places as needed.)

Explanation:

Step1: Differentiate the function \( L = - 37.60+3.71t - 6.32\times10^{-4}t^{3}\)

Using the power rule \(\frac{d}{dt}(x^{n})=nx^{n - 1}\), we get \(\frac{dL}{dt}=3.71-3\times6.32\times10^{-4}t^{2}\)

Step2: Substitute \(t = 15\) into \(\frac{dL}{dt}\)

$$
LATEXBLOCK0
$$

Answer:

\(\frac{dL}{dt}\approx3.28\) mm/week