A balloon is inflated so that its increase in volume is at a constant rate of 10cm^3 s^-1. If its volume is initially 1 cm^3, find its volume after 3 seconds.
Answer: 31 cm^3
Rate of increase:
dV / dt = 10
dV = 10 dt
Integrating both sides:
V = 10 t + C
C = integration constant
Initial condition:
t = 0 , V = 1
V = 10 t + C
1 = 10 ∙ 0 + C
1 = 0 + C
1 = C
C = 1
V = 10 t + C
V = 10 t + 1
After 3 seconds:
V = 10 ∙ 3 + 1
V = 31 cm³
To find the volume of the balloon after 3 seconds, you need to use the formula for the rate of change of volume over time. The given information states that the rate of change of volume is constant at 10 cm^3/s.
The formula for the rate of change of volume is:
ΔV/Δt = 10 cm^3/s
Where:
ΔV is the change in volume
Δt is the change in time (in seconds)
We want to find the volume after 3 seconds, so we can set Δt = 3 s:
ΔV/3s = 10 cm^3/s
To find the change in volume, we can multiply both sides of the equation by 3:
ΔV = 10 cm^3/s * 3 s = 30 cm^3
The change in volume is 30 cm^3.
To find the final volume, we add the change in volume to the initial volume:
Final volume = Initial volume + Change in volume
Final volume = 1 cm^3 + 30 cm^3
Final volume = 31 cm^3
Therefore, the volume of the balloon after 3 seconds is 31 cm^3.