A tank in the shape of an inverted right circular cone contains coffee that is d

Question

A tank in the shape of an inverted right circular cone contains coffee that is draining through the cone's vertex into another tank in the shape of a right circular cylinder. The radius of each tank is 5 meters; the height of each tank is 6 meters. If the coffee level in the cylindrical tank is rising at a rate of 0.15 meter per minute when the coffee level in the conical tank is 2 meters, at exactly what rate is the coffee level in the conical tank changing at this time? Hint: This is really two related rates problems in one.

Explanation / Answer

Here is a method you can follow to solve these problems 1. Diagram Draw a diagram. 2. Goal Find rate (d/dt) of what? 3. Data a. Which quantities are changing? Assign variable names to them. a. Which quantities are constant? b. Which rates are given? Is each rate positive or negative? c. Are there “freeze frame” values? 4. Equation Write an equation relating the variables (e.g. Pythagorean Theorem, area/volume, proportion, trig function). You may need to combine two or more equations. 5. Differentiate Differentiate (d/dt) equation with respect to time. 6. Solve Plug in given values and solve. Useful Formulas Area of a circle: A = pr2 Circumference of a circle: C = 2pr Volume of a sphere: V = 4 3pr3 Surface area of a sphere: A = 4pr2 Volume of a cylinder: V = pr2h Surface area of a cylinder: A = 2prh + 2pr2 Volume of a cone: 1 3pr2h Pythagorean Theorem: a2 + b2 = c2 Pythagorean triples: (3,4,5),(5,12,13)

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