Final answer:
When the water level is 4 cm, water is being poured into the cup at a rate of 100π cm³/sec.
Step-by-step explanation:
To find the rate at which water is being poured into the cup when the water level is 4 cm, we need to find the rate at which the volume of water in the cup is increasing. The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height. Since the radius is half of the diameter, the radius of the cup is 5 cm.
When the water level is 4 cm, we can use similar triangles to find the height of the water in the cup. The ratio of the height of the water to the height of the cup is the same as the ratio of the radius of the water to the radius of the cup. So, the height of the water is (4/10) * 20 = 8 cm.
Now, we can differentiate the volume equation with respect to time t to find the rate at which the volume is changing. Since r and h are changing with time, we need to use the chain rule. The chain rule states that if V is a function of t and r and h are both functions of t, then dV/dt = (∂V/∂r) * (dr/dt) + (∂V/∂h) * (dh/dt).
Plugging in the known values, we get: dV/dt = (1/3) * π(5^2)(4/10) * (0) + (1/3) * π(5^2)(20) * (4/10) = (1/3) * π(5^2)(20) * (4/10) = 100π cm³/sec.
Therefore, when the water level is 4 cm, water is being poured into the cup at a rate of 100π cm³/sec.