Final answer:
The calculation involved analyzing a conical tank with a leak and an inflow, utilizing principles of calculus and related rates to find the change in water height when water is at 200 cm. Similar triangles and the volume of a cone formula were key in determining that change rate.
Step-by-step explanation:
The question deals with evaluating the rate of change in the water height in a leaking and filling conical tank using the concept of related rates in calculus. To solve this problem, we can use the formula for the volume of a cone: V = (1/3) π r^2 h. We're given that the tank leaks at 14 cm³/min and is filled at a rate of 4 cm³/min, which gives us a net rate of -10 cm³/min (loss of water).
Because the cone's dimensions stay similar as it fills or empties, we can use similar triangles to find the relationship between the radius of the water level (r) and the height of the water (h). When the water is at 200 cm high, the radius at the water level is 120 cm. Differentiating both sides of the volume formula with respect to time (t), and substituting known rates and measurements, allows us to solve for the rate at which the height of the water is changing when the water level is 200 cm.