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 X 10^5, R = 1.0 X 10^-2, and dR/dt = 2.0 X 10^-5. (Round your answer to 4 decimal places.)

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 = 1.42 X 10^5, R = 1.0 X 10^-2, and dR/dt = 2.0 X 10^-5. (Round your answer to 4 decimal places.)

S = C(R^2 − r^2)

dS/dt = cm/s

Answer 1

you just want dS/dt, which is just

dS/dt = 2CR dR/dt - 2Cr dr/dt

Now just plug in your values. Note that if r is constant, dr/dt = 0

Answer 2

I can certainly help you with that! To find the rate at which S changes with respect to t, we need to take the derivative of S with respect to t.

First, let's rewrite the equation for S:
S = C(R^2 − r^2)

Now let's take the derivative of S with respect to t. Since S has both R and r in it, we'll need to use the chain rule.

dS/dt = d(C(R^2 − r^2))/dt

Applying the chain rule, we get:

dS/dt = C * d/dt(R^2 − r^2)

The derivative of (R^2 − r^2) with respect to t is:

d/dt(R^2 − r^2) = 2R(dR/dt) − 2r(dr/dt)

Now we can substitute in the given values:

R = 1.0 X 10^-2
dR/dt = 2.0 X 10^-5

Let's also assume that dr/dt is 0, since the rate at which S changes with respect to t is asked for a constant distance r.

Now we can substitute these values back into the derivative equation:

dS/dt = C * (2R(dR/dt) − 2r(dr/dt))
= 1.42 X 10^5 * (2(1.0 X 10^-2)(2.0 X 10^-5) − 2(0))

Simplifying this expression, we get:

dS/dt = 1.42 X 10^5 * (4.0 X 10^-7)
= 5.68 X 10^-2 cm/s

Therefore, the rate at which S changes with respect to t is approximately 0.0568 cm/s.

Answer 3

To find the rate at which S changes with respect to t, we need to find dS/dt.

Given formula:
S = C(R^2 - r^2)

To find dS/dt, we need to differentiate S with respect to t.

Differentiating both sides of the equation with respect to t:

dS/dt = d(C(R^2 - r^2))/dt

Using the chain rule of differentiation, we can differentiate each term separately.

dS/dt = d(C(R^2))/dt - d(r^2)/dt

We know that dR/dt = 2.0 X 10^-5, which represents the rate of change of R with respect to t.

Differentiating C(R^2) with respect to t:
d(C(R^2))/dt = 2C(R)(dR/dt)

Differentiating r^2 with respect to t:
d(r^2)/dt = 2r(dr/dt)

Substituting the given values:
C = 1.42 x 10^5
R = 1.0 x 10^-2
dR/dt = 2.0 x 10^-5

dS/dt = 2C(R)(dR/dt) - 2r(dr/dt)

Substituting the given values:
dS/dt = 2(1.42 x 10^5)(1.0 x 10^-2)(2.0 x 10^-5) - 2r(dr/dt)

Now, we need to substitute the given value of r. However, the problem does not provide the value of r. Please provide the value of r in order to calculate the final answer for dS/dt.

Answer 4

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 find the derivative of S with respect to r:

dS/dr = -2Cr

Next, using the chain rule, we can find dS/dt:

dS/dt = (dS/dr) * (dr/dt)

Now, we know that dR/dt = 2.0 X 10^-5, which is the rate at which the artery is dilating. Since r is constant in this problem, dr/dt = 0.

Therefore, dS/dt = (dS/dr) * (dr/dt) = (dS/dr) * 0 = 0

So, the rate at which S changes with respect to t is 0 cm/s.