Final answer:
To find the rate at which the depth of the coffee is increasing when the cup is half full, you can use the volume formula for a truncated cone and related rates. The rate at which the depth of the coffee is increasing when the cup is half full is approximately 0.343 cm/sec.
Step-by-step explanation:
To find the rate at which the depth of the coffee is increasing when the cup is half full, we need to use related rates. The volume of a truncated cone can be calculated using the formula:
V = (1/3)πh(R^2 + r^2 + Rr), where V is the volume, h is the height, R is the large radius, and r is the small radius. We know that the large diameter is 7.5 cm and the small diameter is 5.5 cm, so the large radius is 3.75 cm and the small radius is 2.75 cm. Let's denote the height of the coffee as h.
Since coffee is poured in at a rate of 15 cm³/sec and we want to find the rate at which the depth of the coffee is increasing when the cup is half full, we need to find dh/dt when V = (1/2)(1/3)πh(3.75^2 + 2.75^2 + 3.75*2.75).
Applying the chain rule, we have:
dV/dt = (1/2)(1/3)π(3.75^2 + 2.75^2 + 3.75*2.75)(dh/dt).
Substituting the given values and solving for dh/dt, we find that the rate at which the depth of the coffee is increasing when the cup is half full is approximately 0.343 cm/sec.