Question

The brain mass of a human fetus during a particular trimester can be accurately estimated from the circumference of the head by ​m(c)equalsStartFraction c cubed Over 100 EndFraction minus StartFraction 1500 Over c EndFraction ​, where​ m(c) is the mass of the brain​ (in grams) and c is the circumference​ (in centimeters) of the head. a. Estimate the brain mass of a fetus that has a head circumference of 20 cm. b. Find the rate of change of the brain mass for a fetus that has a head circumference of 20 cm and interpret your result.

Answer 1

Answer:

a. If c = 20 cm, then the mass of the brain is m = 5 g.

b. At c = 20 cm, the brain's mass is increasing at a rate of 15.75 g/cm.

Explanation:

From the equation

$m\left(c\right) = \frac{c^3}{100}-\frac{1500}{c}$

we have

a. for c = 20 cm

$m\left(20\right)=\frac{20^3}{100}-\frac{1500}{20}=5,$

then the mass is m(20) = 5 g.

b. In order to find the rate of change, first we derivate

$\frac{dm}{dc}=\frac{3c^2}{100}+\frac{1500}{c^2}.$

Evaluated at c = 20 cm, we have

$\frac{dm}{dc}|_{c=20}=\frac{3\times 20^2}{100}+\frac{1500}{20^2}=15.75.$

So, at c = 20 cm, the mass of the brain is increasing at a rate of 15.75 g/cm.