A heart attack victim is given a blood vessel dilator to increase the radii of the blood vessels. After receiving the dilator, the radii of the affected blood vessels increase at about 1% per minute. According to Poiseulle's law, the volume of blood flowing through a vessel and the radius of the vessel are related by the formula V = kr^4 where k is a constant. What will be the percentage rate of increase in the blood flow after the dilator is given?
dV/dt = k d/dt (r)^4 = 4k r^3 dr/dt
(dr/dt)/r = .01
so
dV/dt = 4 k r^3 (.01 r) = .04 k r^4
but we want dV/dt/V times 100
(dV/dt)/V = .04 kr^4/kr^4 = .04
.04 * 100 = 4%
According to Poiseuille's law, the volume of blood flowing through a vessel and the radius of the vessel are related by the formula V = kr^4, where V represents the volume of blood flow and r represents the radius of the blood vessel.
Given that the radii of the affected blood vessels increase at about 1% per minute, we can calculate the percentage rate of increase in the blood flow.
Let's assume the initial radius of the blood vessel is r0 and the rate of increase is 1% per minute. This means that after one minute, the radius of the vessel will increase to (r0 + 0.01r0) = 1.01r0.
To find the percentage rate of increase in blood flow, we need to compare the initial flow (V0) to the new flow (V1).
Using the formula V = kr^4, we can calculate the initial and new flows:
V0 = k(r0)^4
V1 = k(1.01r0)^4
Now we can find the percentage rate of increase in blood flow:
Percentage rate of increase in blood flow = ((V1 - V0) / V0) * 100
Substituting the values we have:
Percentage rate of increase in blood flow = ((k(1.01r0)^4 - k(r0)^4) / k(r0)^4) * 100
Percentage rate of increase in blood flow = (1.04 - 1) * 100
Percentage rate of increase in blood flow = 4%
Therefore, the percentage rate of increase in blood flow after receiving the dilator will be 4%.
To find the percentage rate of increase in blood flow after the dilator is given, we need to determine how the change in the radius affects the volume of blood flowing through the vessel.
According to Poiseuille's Law, the volume of blood flow is directly proportional to the fourth power of the radius. So, if the radius increases by 1%, we can substitute this value into the formula and calculate the percentage change in blood flow.
Let's assume the initial radius is denoted as "r" and the initial volume of blood flow as "V". After 1% increase in the radius, the new radius would be 1.01r (since 1% of r is 0.01r). Therefore, the new volume of blood flow, V', can be calculated as follows:
V' = k(1.01r)^4
Expanding the equation:
V' = k(1.01)^4r^4
V' = 1.0406kr^4
Now, let's calculate the percentage increase in blood flow (ΔV%) by subtracting the initial volume (V) from the new volume (V') and then dividing by the initial volume (V):
ΔV% = ((V' - V) / V) * 100
Substituting the calculated values:
ΔV% = ((1.0406kr^4 - kr^4) / kr^4) * 100
ΔV% = ((0.0406kr^4) / kr^4) * 100
ΔV% = 4.06%
Therefore, the percentage rate of increase in the blood flow after the dilator is given is approximately 4.06%.