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 = 1.42 105, R = 1.0 10-2, and dR/dt = 2.0 10-5. (Round your answer to 4 decimal places.)
S = C(R2 − r2)
dS/dt = cm/s
To find the rate at which S changes with respect to t, we need to differentiate the equation S = C(R^2 - r^2) with respect to t.
First, let's rewrite the equation in a simpler form by expanding the square terms, R^2 and r^2:
S = C(R^2 - r^2)
S = C(R + r)(R - r)
Now, let's differentiate both sides of the equation with respect to t using the product rule for differentiation:
dS/dt = d/dt [C(R + r)(R - r)]
Using the product rule, we differentiate each term separately:
dS/dt = C [(d/dt)(R + r)(R - r) + (R + r)(d/dt)(R - r)]
Now, we need to find the values of d(R + r)/dt and d(R - r)/dt. Given that dR/dt = 2.0 * 10^-5, we can substitute this value into the derivatives:
d(R + r)/dt = dR/dt + dr/dt = 2.0 * 10^-5 + 0 (since dr/dt is not given)
d(R - r)/dt = dR/dt - dr/dt = 2.0 * 10^-5 - 0 (since dr/dt is not given)
Substituting these values back into the equation:
dS/dt = C [(2.0 * 10^-5 + 0)(R - r) + (R + r)(2.0 * 10^-5 - 0)]
Since r is a constant distance and not changing with respect to time, the derivative dr/dt is zero. Therefore, we simplify the equation further:
dS/dt = C [2.0 * 10^-5(R - r) + (R + r)(2.0 * 10^-5)]
Now, substitute the given values C = 1.42 * 10^5, R = 1.0 * 10^-2, and dR/dt = 2.0 * 10^-5 into the equation:
dS/dt = 1.42 * 10^5 [2.0 * 10^-5(1.0 * 10^-2 - r) + (1.0 * 10^-2 + r)(2.0 * 10^-5)]
Simplifying this equation will give you the rate at which S changes with respect to t.