Samuel bought a cement mixer for $54,205. The value of the cement mixer depreciated at a constant rate per year. The table shows the value of the cement mixer after the first and second years:
| Year | 1 | 2 |
|------|-------------|-------------|
| Value (in dollars) | $47,158.35 | $41,027.76 |
Which function best represents the value of the cement mixer after [tex]$t$[/tex] years?
A. [tex]$f(t)=47,158.35(0.87)^t$[/tex]
B. [tex]$f(t)=54,205(0.13)^t$[/tex]
C. [tex]$f(t)=47,158.35(0.13)^t$[/tex]
D. [tex]$f(t)=54,205(0.87)^t$[/tex]
Answer (2)
The problem describes an exponential decay situation.
Calculate the depreciation rate: r a t e = 54205 47158.35 = 0.87 .
Formulate the exponential decay function: f ( t ) = 54205 ( 0.87 ) t .
The function that best represents the value of the cement mixer after t years is f ( t ) = 54 , 205 ( 0.87 ) t .
Explanation
Understanding the Problem We are given the initial value of a cement mixer and its value after the first and second years. We are told that the value depreciates at a constant rate per year, and we need to find the function that represents the value of the cement mixer after t years.
General Form of Exponential Decay The general form of an exponential decay function is given by f ( t ) = I ni t ia l Va l u e ∗ ( r a t e ) t , where I ni t ia l Va l u e is the value at t = 0 and r a t e is the depreciation rate per year.
Calculating the Depreciation Rate The initial value of the cement mixer is $54 , 205 . We need to find the rate of depreciation. We can calculate the rate by dividing the value after 1 year by the initial value: r a t e = I ni t ia l Va l u e Va l u e ye a r 1 = 54205 47158.35 = 0.87 . Alternatively, we can calculate the rate by dividing the value after 2 years by the value after 1 year: r a t e = Va l u e ye a r 1 Va l u e ye a r 2 = 47158.35 41027.76 = 0.87 .
Finding the Function Now we can write the function that represents the value of the cement mixer after t years: f ( t ) = 54205 ∗ ( 0.87 ) t .
Final Answer Comparing this function with the given options, we see that the correct function is f ( t ) = 54 , 205 ( 0.87 ) t .
Examples
Understanding depreciation is crucial in business and personal finance. For instance, when a company buys equipment, like our cement mixer, its value decreases over time. This decrease, known as depreciation, affects the company's taxes and financial statements. By using the formula we found, f ( t ) = 54205 ∗ ( 0.87 ) t , the company can predict the mixer's value in future years. This helps in planning for replacements and understanding the true cost of owning the equipment. Similarly, individuals can use depreciation calculations to understand the decreasing value of their assets, like cars, for insurance and financial planning purposes.
The function that represents the value of Samuel's cement mixer after t years is f ( t ) = 54 , 205 ( 0.87 ) t . This indicates that the value of the mixer depreciates by a factor of 0.87 each year. The correct multiple choice option is D.
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