The speed S of blood that is r centimeters from the center of an artery is given below, where C is a constant, R is the radius of the artery, and S is measured in centimeters per second. Suppose a drug is administered and the artery begins to dilate at a rate of dR/dt. At a constant distance r, find the rate at which S changes with respect to t for C = 1.54 105, R = 1.3 10-2, and dR/dt = 1.0 10-5.

S = C(R^2 − r^2)

dS/dt =______cm/s

Question

The speed S of blood that is r centimeters from the center of an artery is given below, where C is a constant, R is the radius of the artery, and S is measured in centimeters per second. Suppose a drug is administered and the artery begins to dilate at a rate of dR/dt. At a constant distance r, find the rate at which S changes with respect to t for C = 1.54 105, R = 1.3 10-2, and dR/dt = 1.0 10-5. 

S = C(R^2 − r^2) 

dS/dt =______cm/s

anonymous · Answer

$$S = C*R ^{2} - C*r ^{2}$$
take derivative with respect to t on both sides, using chain rule
$$\frac{ dS}{ dt } = C*(2*R)*(\frac{ dR }{ dt }) - C*(2*R)*(\frac{ dr }{ dt })$$
little r does not depend on t, so $$\frac{ dr }{ dt } = 0$$
$$\frac{ dS }{ dt } = 2CR*\frac{ dR }{ dt } - 2CR*(0)$$
plug in for variables