Answer :
Answer:
t = 1064 s
Explanation:
This exercise will work with the equations of fluid mechanics, let's start by calculating the filling flow rate
Q = A v
Area of the circle is
r = d / 2 = 0.50 m
A = π R²
A = π 0.50²
A = 0.785 m²
Q = 0.785 3
Q = 2,356 m · / s
Let's calculate the initial and final volume
Ro = 5/2 m = 2.5 m
Rf = 17/2 m = 8.5 m
V = 4/3 π r³
Vo = 4/3 π 2.5³
Vo = 65.45 m³
Vf = 4/3 π 8.5³
Vf = 2572.44 m³
The difference is the volume to be filled
ΔV = Vf - Vo
ΔV = 2572.44 - 65.45
ΔV = 2506.99 m3
As we have the flow, let's calculate the time
Q = V / t
t = V / Q
t = 2506.99 / 2,356
t = 1064 s
To solve the problem we must know about the concept of flow rate.
What is the Flow rate?
Flow rate is the rate at which a fluid flows into a pipe. It is given by the formula,
[tex]\dot Q = v \times A[/tex]
Where Q is the flow rate, v is the velocity of the fluid, and A is the cross-sectional area of the pipe.
The time is taken to change the volume of the ballon is 1064 sec.
To know the time taken by the air to fill the balloon we need to calculate the amount of air that is needed to be pumped into the balloon, therefore we will calculate the change in the volume of the ballons,
The volume of the ballon = [tex]\dfrac{4}{3}\pi r^3[/tex]
The volume of air that is needed to be filled
[tex]=\dfrac{4}{3}\pi (R^3 - r^3)\\\\=\dfrac{4}{3}\pi [(\dfrac{17}{2})^3 - (\dfrac{5}{2})^3]\\\\=2506.99\rm\ m^3[/tex]
Now, using the flow rate formula, we will calculate the flow rate through which the air is been filled in the ballon,
[tex]\dot Q = v \times A[/tex]
[tex]\dot Q = 3 \times (\pi r^2)\\\\\dot Q = 3 \times (\pi 0.5^2)\\\\\dot Q = 2.3562\ m^3 /s[/tex]
What is the time taken to fill the balloon?
The time is taken to fill the ballon,
[tex]V = \dot Q \times t[/tex]
[tex]2506.99 = 2.3562 \times t\\\\t = 1063.99\rm\ sec[/tex]
Hence, the time is taken to change the volume of the ballon is 1064 sec.
Learn more about Flow rate:
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