Fluid is flowing in a tube that has a radius of 3 centimeters. Water is flowing through a circular cross section at a rate of (9-r^2) cm/s, where r is the distance from the center of the cross section. What is the total amount of water that flows through the cross section in 4 seconds?
ok, to get the flow, you integrate area*flow rate through that area.
so dArea= dr*2PI
dVolume= flowrate*dArea
Volume= INT 9r^2*2PI*dr=18PI r^3/3 from r=0 to r=3
volumeflowrate= 6PI *27 cm^3/sec
total amount H2O: volumeflowrate*time
through every ring of radius r and thickness dr and amount per second flows equal to 2 pi r v dr
or
dq = 2 pi (9 r - r^3) dr
integrate that from 0 to 3 and then multiply by 4 seconds
18 pi (r^2/2) - 2 pi (r^4/4)
9 pi (9) - (pi/2)(81)
Well, let's first find the area of the cross section. The area of a circle is given by the formula A = πr^2, where r is the radius of the circle. In this case, the radius is 3 cm, so the area of the cross section is A = π(3^2) = 9π cm^2.
Now, we can find the total amount of water that flows through the cross section in 4 seconds by integrating the flow rate function over the cross section. The flow rate at a distance r from the center is given by (9 - r^2) cm/s. To find the total amount of water that flows through the cross section, we need to integrate this flow rate function from r = 0 to r = 3 over the 4-second period.
Integrating (9 - r^2) from r = 0 to r = 3 gives:
∫[(9 - r^2)] dr = [9r - (r^3)/3] evaluated from 0 to 3
Plugging in the limits:
[9(3) - (3^3)/3] - [9(0) - (0^3)/3]
= [27 - 9] - [0 - 0]
= 18 cm^3
So, the total amount of water that flows through the cross section in 4 seconds is 18 cm^3.
To find the total amount of water that flows through the cross section in 4 seconds, we need to calculate the volume of water that flows through the cross section per second, and then multiply it by the total time.
To do this, we first need to find the area of the circular cross section at each distance "r" from the center. The formula for the area of a circle is A = π * r^2.
Given that the radius of the tube is 3 centimeters, the area at each distance "r" is A = π * (3^2 - r^2) square centimeters.
Now, we need to find the volume of water that flows through each ring of the cross section per second. The volume is given by the formula V = A * v, where "v" is the velocity of the water at each distance "r".
Given that the velocity at each distance "r" is (9 - r^2) cm/s, the volume of water flowing through each ring per second is V = (π * (3^2 - r^2)) * (9 - r^2) cubic centimeters.
To find the total amount of water that flows through the cross section in 4 seconds, we need to integrate this volume over the range of distances "r" from 0 to 3, and then multiply by 4.
The integral is given by the formula: ∫[0,3] (π * (3^2 - r^2)) * (9 - r^2) dr.
Evaluating this integral will give us the total volume of water that flows through the cross section in 4 seconds.
To find the total amount of water that flows through the cross section in 4 seconds, we need to calculate the volume of water that passes through the cross section per second, and then multiply it by the time duration.
Step 1: Calculate the area of the circular cross section
The area of a circle can be calculated using the formula A = π * r^2, where A is the area and r is the radius. Since the radius is given as 3 centimeters, we can substitute it into the formula:
A = π * (3 cm)^2
A = π * 9 cm^2
Step 2: Calculate the volume of water passing through the cross section per second
The rate at which water is flowing through the cross section is given by (9 - r^2) cm/s. In this case, the velocity is not constant and depends on the distance from the center of the cross section, r. To calculate the volume flow rate, we multiply the area of the cross section by the velocity:
Volume flow rate = A * velocity
Volume flow rate = π * 9 cm^2 * (9 - r^2) cm/s
Step 3: Calculate the total amount of water that flows through the cross section in 4 seconds
We have the volume flow rate in cm^3/s. To find the total volume of water in 4 seconds, we multiply the volume flow rate by the time duration:
Total volume = Volume flow rate * time
Total volume = (π * 9 cm^2 * (9 - r^2) cm/s) * 4 s
Now you can substitute the given values and solve the equation to find the total amount of water that flows through the cross section in 4 seconds.