Answer:
The rate at which the radius is decreasing when the radius is 6 cm is approximately 3.316 × 10⁻³ cm/s
Step-by-step explanation:
The rate at which air is being lost from the balloon = 3/2 cm³/s
The rate at which the radius is decreasing when the radius is 6 cm long is given as follows;
The rate at which air is being lost from the balloon = dV/dt = 3/2 cm³/s
dV/dt = dV/dr × dr/dt
Where;
dr/dt = The rate at which the radius is decreasing
dV/dr = d(4/3×π×r³)/dr = 4·π·r²
Therefore, we have;
dr/dt = (dV//dt)/(dV/dr) = (3/2 cm³/s)/(4·π·r²)
dr/dt = (3/2 cm³/s)/(4·π·r²)
When r = 6 cm, we have;
dr/dt = (3/2 cm³/s)/(4 × π × (6 cm)²) ≈ 3.316 × 10⁻³ cm/s
Therefore, the rate at which the radius is decreasing, dr/dt, when the radius is 6 cm long ≈ 3.316 × 10⁻³ cm/s.
