You and your friends are designing and constructing a cone-shaped funnel. Whatever you use this funnel for is up to you. You decide that the height of the funnel will be 6 inches. (a) What does the radius of the base of the funnel need to be if you want the volume of it to be 69 inches cubed? (V = 1/3 πr 2h, where r is the radius of the circular base, and h is the height.) (b) You finally construct it, with the correct height and radius, and you pour the liquid into it to test it out. Again, the choice of liquid and the reason you are doing this is up to you. In measuring the effectiveness of the funnel, you determine that as the liquid flows out of the bottom, the radius of the liquid as it’s in the funnel is decreasing at a rate of 1 inch per second, and the height is decreasing at a rate of 1.75 inches per second. Find the rate at which the volume of the liquid (in inches cubed per second) is decreasing. (c) How many seconds will it take for the funnel to be empty at this rate?
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
You and your friends are designing and constructing a cone-shaped funnel. Whatever you use this funnel for is up to you. You decide that the height of the funnel will be 6 inches.
(a) What does the radius of the base of the funnel need to be if you want the volume of it to be 69 inches cubed? (V = 1/3 πr 2h, where r is the radius of the circular base, and h is the height.)
(b) You finally construct it, with the correct height and radius, and you pour the liquid into it to test it out. Again, the choice of liquid and the reason you are doing this is up to you. In measuring the effectiveness of the funnel, you determine that as the liquid flows out of the bottom, the radius of the liquid as it’s in the funnel is decreasing at a rate of 1 inch per second, and the height is decreasing at a rate of 1.75 inches per second. Find the rate at which the volume of the liquid (in inches cubed per second) is decreasing.
(c) How many seconds will it take for the funnel to be empty at this rate?
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