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 } \\)

calculate the rate of growth at t = 15 weeks.

\\( \frac { d l } { d t } = 3.28 \\) mm/week
(round to two decimal places as needed.)

calculate the rate of growth at t = 25 weeks.

\\( \frac { d l } { d t } = 2.52 \\) mm/week
(round to two decimal places as needed.)

Explanation:

Step1: Differentiate \(L\) with respect to \(t\)

Given \(L=-37.60 + 3.71t-6.32\times10^{-4}t^{3}\).
Using the power rule \(\frac{d}{dt}(t^{n})=nt^{n - 1}\) and \(\frac{d}{dt}(C)=0\) (where \(C\) is a constant).
\(\frac{dL}{dt}=\frac{d}{dt}(-37.60)+\frac{d}{dt}(3.71t)-\frac{d}{dt}(6.32\times 10^{-4}t^{3})\)
\(\frac{dL}{dt}=0 + 3.71-3\times6.32\times10^{-4}t^{2}\)
\(\frac{dL}{dt}=3.71-1.896\times10^{-3}t^{2}\)

Step2: Calculate \(\frac{dL}{dt}\) at \(t = 10\)

Substitute \(t = 10\) into \(\frac{dL}{dt}=3.71-1.896\times10^{-3}t^{2}\)
\(\frac{dL}{dt}\mid_{t = 10}=3.71-1.896\times10^{-3}\times(10)^{2}\)
\(=3.71-1.896\times0.1\)
\(=3.71 - 0.1896\)
\(=3.5204\approx3.52\)

Step3: Calculate \(\frac{dL}{dt}\) at \(t = 15\)

Substitute \(t = 15\) into \(\frac{dL}{dt}=3.71-1.896\times10^{-3}t^{2}\)
\(\frac{dL}{dt}\mid_{t = 15}=3.71-1.896\times10^{-3}\times(15)^{2}\)
\(=3.71-1.896\times0.225\)
\(=3.71-0.4266\)
\(=3.2834\approx3.28\)

Step4: Calculate \(\frac{dL}{dt}\) at \(t = 25\)

Substitute \(t = 25\) into \(\frac{dL}{dt}=3.71-1.896\times10^{-3}t^{2}\)
\(\frac{dL}{dt}\mid_{t = 25}=3.71-1.896\times10^{-3}\times(25)^{2}\)
\(=3.71-1.896\times0.625\)
\(=3.71 - 1.185\)
\(=2.525\approx2.53\)

Answer:

When \(t = 10\), \(\frac{dL}{dt}\approx3.52\) mm/week; when \(t = 15\), \(\frac{dL}{dt}\approx3.28\) mm/week; when \(t = 25\), \(\frac{dL}{dt}\approx2.53\) mm/week.
Since \(3.52>3.28>2.53\), the rate of growth of the fetus decreases as it ages.