A room has a volume of 120 m^3. An air-conditioning system is used to replace the air in the room every twenty minutes, using ducts that have a square cross ection. Assuming that air can be treated as an incompressible fluid, calculate the length of a side of the square if the air speed within the ducts is 3.0m/s and 5.0m/s.

Please and thank you.

Use the formula V=Avt

a)120m^3=A(3.0m/s)(1200s)
solve for A
and take the square root of the result will give you 0.18m
b)12m^3=A(5.0m/s)(1200s)
solve for A
and take the square root of the result
will give you 0.14m

Well, let's have a blast solving this problem!

To calculate the length of a side of the square duct, we'll need to first find the volume of air that passes through it in twenty minutes. Then we can use the volume and the air speed to determine the length of the duct.

Now, the volume of air that needs to be replaced in twenty minutes is the same as the volume of the room, which is 120 m^3.

Let's start by converting the time into seconds, since the air speed is given in meters per second and it'll help us later. Twenty minutes is equal to 20 * 60 = 1200 seconds.

Now, to find the volume of air passing through the duct each second, we divide the total volume of 120 m^3 by 1200 seconds. This gives us a flow rate of 0.1 m^3/s.

If the air speed within the duct is 3.0 m/s, we divide the flow rate by the air speed to find the cross-sectional area of the duct. So, 0.1 m^3/s divided by 3.0 m/s equals 0.033 m^2.

To find the length of a side of the square duct, we need to calculate the square root of the cross-sectional area. The square root of 0.033 m^2 is approximately 0.182 m.

If the air speed within the duct is 5.0 m/s, we follow the same steps. The flow rate of 0.1 m^3/s divided by 5.0 m/s gives us a cross-sectional area of 0.02 m^2. Taking the square root of 0.02 m^2, we get approximately 0.141 m.

So, if the air speed is 3.0 m/s, the length of a side of the square duct is about 0.182 m. And if the air speed is 5.0 m/s, the length of a side of the square duct is around 0.141 m.

I hope this answer blows you away with its accuracy and humor!

To find the length of a side of the square duct, we first need to calculate the volume flow rate. The volume flow rate can be calculated using the formula Q = A * V, where Q is the volume flow rate, A is the cross-sectional area of the duct, and V is the air speed.

1. Calculate the volume flow rate (Q) using the given air speed (V) and room volume (120 m^3):
Q = 120 m^3 / 20 min
Note: We need to convert the time from minutes to seconds.
Q = 120 m^3 / (20 min * 60 s/min)
Q = 0.1 m^3/s

2. To find the cross-sectional area (A), we rearrange the formula Q = A * V:
A = Q / V

Now, we can find the length of a side of the square duct using the calculated cross-sectional area (A).

If the air speed (V) is 3.0 m/s:
3. Calculate the cross-sectional area (A):
A = 0.1 m^3/s / 3.0 m/s
A = 0.0333 m^2

4. Find the length of a side of the square duct (L) using the formula for the area of a square:
L = sqrt(A)
L = sqrt(0.0333 m^2)
L ≈ 0.182 m

Therefore, if the air speed within the duct is 3.0 m/s, the length of a side of the square duct should be approximately 0.182 m.

If the air speed (V) is 5.0 m/s, the calculations are as follows:

3. Calculate the cross-sectional area (A):
A = 0.1 m^3/s / 5.0 m/s
A = 0.02 m^2

4. Find the length of a side of the square duct (L) using the formula for the area of a square:
L = sqrt(A)
L = sqrt(0.02 m^2)
L ≈ 0.141 m

Therefore, if the air speed within the duct is 5.0 m/s, the length of a side of the square duct should be approximately 0.141 m.

To calculate the length of a side of the square duct, we need to use the equation for volume flow rate. The volume flow rate (Q) is equal to the product of cross-sectional area (A) and the velocity (v) of the fluid.

In this case, the volume flow rate (Q) is equal to the volume of the room (V) divided by the time taken to replace the air in the room (t). Therefore, we can write the equation as follows:

Q = V / t

Now, let's calculate the volume flow rate for both scenarios where the air speeds are 3.0 m/s and 5.0 m/s.

1) Air speed = 3.0 m/s:
In this case, the volume flow rate (Q) is equal to the volume of the room divided by the time taken to replace the air in the room.

Q = V / t = 120 m^3 / 20 min = 6 m^3/min

Since we are given the air speed (v) as 3.0 m/s, we can use the equation for volume flow rate to determine the cross-sectional area (A) of the duct.

Q = A * v

Substituting the values, we have:

6 m^3/min = A * 3.0 m/s

Solving for A, we can rearrange the equation as follows:

A = Q / v = 6 m^3/min / 3.0 m/s = 2 m^2/s

Since we are dealing with a square duct, the cross-sectional area (A) is equal to the length of one side squared (s^2).

A = s^2

Substituting the value of A, we can solve for s:

2 m^2/s = s^2

Taking the square root of both sides, we get:

s = √(2 m^2/s) ≈ 1.41 m

Therefore, the length of one side of the square duct when the air speed is 3.0 m/s is approximately 1.41 m.

2) Air speed = 5.0 m/s:
Using the same process, we can calculate the length of one side of the square duct when the air speed is 5.0 m/s.

Q = V / t = 120 m^3 / 20 min = 6 m^3/min

A = Q / v = 6 m^3/min / 5.0 m/s = 1.2 m^2/s

s = √(1.2 m^2/s) ≈ 1.10 m

Therefore, the length of one side of the square duct when the air speed is 5.0 m/s is approximately 1.10 m.

In summary:
- When the air speed is 3.0 m/s, the length of one side of the square duct is approximately 1.41 m.
- When the air speed is 5.0 m/s, the length of one side of the square duct is approximately 1.10 m.