A worker is reviewing his pay increases over the past several years. The table below shows the hourly wage W, in dollars, that he earned as a function of time t, measured in years since the beginning of 2000. Time t Wage W 1 15.80 2 15.95 3 16.10 4 16.25 (a) By calculating ratios, show that the data in this table are exponential. (Round your answer to two decimal places.) The ratios of successive terms are always , which shows that the data is exponential. (b) What is the yearly growth factor for the data? (Round your answer to two decimal places.) (c) The worker can't remember what hourly wage he earned at the beginning of 2000. Assuming that W is indeed an exponential function, determine what that hourly wage was. (Round your answer to the nearest cent.) $ (d) Find a formula giving an exponential model for W as a function of t. (Round the parameters to two decimal places.) W(t) = 31.29 × 1.04t W(t) = 19.55 × 1.91t W(t) = 7.82 × 1.08t W(t) = 11.73 × 0.99t W(t) = 15.64 × 1.01t (e) What percentage raise did the worker receive each year? (Use the model found in part (d).) % (f) Given that prices increased by 26% over the decade of the 2000s, use your model to determine whether the worker's wage increases kept pace with inflation. Yes, the wage increases kept pace with inflation rate.No, the wage increases were lower than inflation 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.
A worker is reviewing his pay increases over the past several years. The table below shows the hourly wage W, in dollars, that he earned as a function of time t, measured in years since the beginning of 2000.
| Time t | Wage W |
|---|---|
| 1 | 15.80 |
| 2 | 15.95 |
| 3 | 16.10 |
| 4 | 16.25 |
(b) What is the yearly growth factor for the data? (Round your answer to two decimal places.)
(c) The worker can't remember what hourly wage he earned at the beginning of 2000. Assuming that W is indeed an exponential function, determine what that hourly wage was. (Round your answer to the nearest cent.)
$
(d) Find a formula giving an exponential model for W as a function of t. (Round the parameters to two decimal places.)
(e) What percentage raise did the worker receive each year? (Use the model found in part (d).)
%
(f) Given that prices increased by 26% over the decade of the 2000s, use your model to determine whether the worker's wage increases kept pace with inflation.
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