Question

a conical water tank with vertex down has a radius of 15 feet at the top and is 21 feet high. if water flows into the tank at a rate of 20 ft3/min , how fast is the depth of the water increasing when the water is 14 feet deep?

Answer 1

Using related rates in calculus and the volume formula for a cone, we can determine the rate of increase in water depth in a conical tank when given the flow rate and the depth at a specific moment.

Step-by-step explanation:

The question involves applying the concepts of related rates in calculus to find how fast the depth of the water in a conical tank increases. First, we'll use the volume formula for a cone, V = (1/3)πr^2h, where V is volume, r is the radius, and h is the height. Since the tank's radius and height have a constant ratio because of its conical shape, we have r/h = 15/21. This allows us to express r in terms of h as r = (15/21)h.

Given the rate of flow into the tank (dV/dt) is 20 ft^3/min, we want to determine the rate at which the water level (dh/dt) is rising when h = 14 feet. Using the chain rule, the relation between dV/dt and dh/dt is found by differentiating the volume formula with respect to t. Substituting the given values and solving for dh/dt will give us the desired rate at which the water depth increases.