The brain mass of a fetus can be estimated using the total mass of the fetus by the function b=0.22m^0.87, where m is the mass of the fetus (in grams) and b is the brain mass (in grams). Suppose the brain mass of a 28-g fetus is changing at a rate of 0.29 g per day. Use this to estimate the rate of change of the total mass of the fetus, dm/dt.
To find the rate of change of the total mass of the fetus, we need to use the chain rule.
Given:
b = 0.22m^0.87
t = time in days
We are given that db/dt = 0.29 g/day. We are looking for dm/dt when m = 28g.
Taking the derivative of the function with respect to t:
db/dt = d(0.22m^0.87)/dt
db/dt = 0.22 * 0.87 * m^(-0.13) * dm/dt
Substitute b = 0.22(28)^0.87 and db/dt = 0.29 into the equation:
0.29 = 0.22 * 0.87 * (28)^(-0.13) * dm/dt
Solve for dm/dt:
dm/dt = 0.29 / (0.22 * 0.87 * (28)^(-0.13))
dm/dt = 0.29 / (0.1914 * 0.6377)
dm/dt ≈ 2.24 g/day
Therefore, the rate of change of the total mass of the fetus is approximately 2.24 grams per day.