A restaurant purchased kitchen equipment on January 1, 2017. On January 1, 2019, the value of the equipment was $14,450. The value after that date was modeled as follows:
[tex]V(t)=14,450 e^{-0.159 n}[/tex]
a) What is the rate of change in the value of the equipment on January 1, 2019?
b) What was the original value of the equipment on January 1, 2017?
a) The rate of change in the value of the equipment on January 1, 2019 was $\square$ dollars per year.
(Type an integer or decimal rounded to two decimal places as needed.)
Answer (1)
Calculate the rate of change on January 1, 2019 by finding the derivative of the function V ( n ) and evaluating it at n = 0 , resulting in approximately − 2297.55 .
Determine the original value on January 1, 2017 by using the given value on January 1, 2019 and solving for V 0 in the equation 14450 = V 0 e − 0.159 ( 2 ) , which gives approximately 19859.74 .
Explanation
Problem Analysis We are given the function V ( t ) = 14450 e − 0.159 n , where V ( t ) is the value of the equipment at time t , and n is the number of years after January 1, 2019. We need to find the rate of change of the value of the equipment on January 1, 2019, and the original value of the equipment on January 1, 2017.
Finding the Rate of Change a) To find the rate of change of the value of the equipment on January 1, 2019, we need to find the derivative of V ( n ) with respect to n and evaluate it at n = 0 . The derivative of V ( n ) is:
V ′ ( n ) = d n d ( 14450 e − 0.159 n ) = 14450 ( − 0.159 ) e − 0.159 n
Now, we evaluate V ′ ( n ) at n = 0 :
V ′ ( 0 ) = 14450 ( − 0.159 ) e − 0.159 ( 0 ) = 14450 ( − 0.159 ) e 0 = 14450 ( − 0.159 ) = − 2297.55
So, the rate of change in the value of the equipment on January 1, 2019, is -$2297.55 per year.
Finding the Original Value b) To find the original value of the equipment on January 1, 2017, we need to find the value of the equipment when t = 0 . Since n is the number of years after January 1, 2019, and t is the number of years after January 1, 2017, we have n = t − 2 . Thus, when t = 0 , n = 0 − 2 = − 2 .
We know that on January 1, 2019, the value was $14450. So, we need to find the value on January 1, 2017. Let V 0 be the original value on January 1, 2017. Then:
14450 = V 0 e − 0.159 ( 2 )
Solving for V 0 :
V 0 = e − 0.159 ( 2 ) 14450 = e − 0.318 14450 ≈ 0.7274 14450 ≈ 19859.74
So, the original value of the equipment on January 1, 2017, was approximately $19859.74.
Final Answer a) The rate of change in the value of the equipment on January 1, 2019 was − 2297.55 dollars per year. b) The original value of the equipment on January 1, 2017 was 19859.74 dollars.
Examples
Understanding depreciation is crucial in business for tax purposes and asset valuation. For instance, a company might use the declining balance method, similar to the exponential decay in this problem, to write off the value of equipment over its useful life. This allows them to accurately reflect the asset's value on their balance sheet and claim depreciation expenses, reducing their taxable income. Proper depreciation accounting helps businesses make informed decisions about when to replace assets and manage their financial health effectively.
