The mass, M, of a child can be approximated based on the height, H, of the child. The height of the child can be projected based on the child's age, A.
a) State the chain rule for the derivative of the mass with respect to age (ie. find dM/dA)
b) Suppose that an allometric model of the mass in kg and height in m of a 6 to 10 year old male child is given by M(H)= 16.5H^2. Further suppose that the height of 6 to 10 year old male child can be approximated by the linear model H(A) = 0.065A + 0.68 where the age is given in years. Find dM/dA.
I am having trouble with coming up with the derivative and knowing where to put specific things in specific places. Please help.
To find the derivative of the mass, M, with respect to age, A (dM/dA), we can use the chain rule. The chain rule states that if we have a composition of functions, such as M(H(A)), and we want to find the derivative with respect to an inner variable (in this case, A), we need to consider the derivatives of the outer and inner functions.
a) Applying the chain rule, the derivative of M with respect to A (dM/dA) can be found using the following formula:
dM/dA = (dM/dH) * (dH/dA)
Here, dM/dH represents the derivative of the mass M with respect to the height H, and dH/dA represents the derivative of the height H with respect to the age A.
b) In this case, the given mass-height relation is M(H) = 16.5H^2. The height-age relation is H(A) = 0.065A + 0.68.
To find dM/dA, we first need to find dM/dH and dH/dA separately.
Applying the power rule, we find:
dM/dH = 2 * 16.5 * H^(2-1)
= 33H
Differentiating the height-age relation:
dH/dA = 0.065
Now, we can substitute these derivatives into the chain rule formula:
dM/dA = (dM/dH) * (dH/dA)
= 33H * 0.065
To determine the specific value of dM/dA, we need to know the value of H(A) at a particular age. For example, if we want to find dM/dA at age A=6, we can substitute this value into the height-age relation H(A):
H(6) = 0.065 * 6 + 0.68
= 1.15
Now, substituting the value of H into the derivative expression:
dM/dA = 33 * 1.15 * 0.065
Evaluating this expression will give you the value of dM/dA at the specific age given.