Question

Air is being lost from a spherical balloon at a constant rate of 3/2cm3s-1. Find the rate at which the radius is decreasing at the instant when the radius is 6 cm long.

Answer 1

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.