5 An oil tanker crashes off the coast of Florida at 12: 00 pm (t oil spreads out over the surface of the ocean in a circle, with the radius at time t given in kilometers by r(t). The total volume of oil spilled as a function of the radius is given by V(r) in liters. Assume V (r) and r(t) are invertible functions. For each of the following quantities, write out an expression that represents each quantity. Your expressions should involve only the functions V, V-1, r and r-1 evaluated at different inputs. 0 hours), causing oil to leak out. The (a) The volume of the oil spill at 2 o'clock: (b) The time when the oil spill reached a radius of 5 km: (c) The time when the volume of the oil spill reached 20,000 L: (d) The change in the radius as the volume increases from 5,000L to 10,000L: (e) The amount of time it will take for the radius to grow from 5 km to 6 km:
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.
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