a restaurant purchased kitchen equipment on january 1, 2017. on january 1, 2019, the value of the equipment…

a restaurant purchased kitchen equipment on january 1, 2017. on january 1, 2019, the value of the equipment was $14,650. the value after that date was modeled as follows. v(t)=14650e^{-0.168t} 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 - 2461.20 dollars per year (type an integer or decimal rounded to two decimal places as needed.) b) the original value of the equipment on january 1, 2017 was $ (type an integer or decimal rounded to two decimal places as needed.)

a restaurant purchased kitchen equipment on january 1, 2017. on january 1, 2019, the value of the equipment was $14,650. the value after that date was modeled as follows. v(t)=14650e^{-0.168t} 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 - 2461.20 dollars per year (type an integer or decimal rounded to two decimal places as needed.) b) the original value of the equipment on january 1, 2017 was $ (type an integer or decimal rounded to two decimal places as needed.)

Answer

Explanation:

Step1: Recall the formula for the derivative of an exponential - function

The derivative of $y = ae^{bx}$ is $y^\prime=abe^{bx}$, where $a = 14650$ and $b=- 0.168$ in the function $V(t)=14650e^{-0.168t}$. So, $V^\prime(t)=14650\times(-0.168)e^{-0.168t}=-2461.2e^{-0.168t}$.

Step2: Evaluate the derivative at $t = 0$ (January 1, 2019)

When $t = 0$, $V^\prime(0)=-2461.2e^{-0.168\times0}=-2461.2$ (which is consistent with the given answer for part a).

Step3: Find the original value (January 1, 2017)

The function $V(t)=14650e^{-0.168t}$ gives the value of the equipment starting from January 1, 2019. Let's assume that from January 1, 2017 to January 1, 2019, $t = 0$ corresponds to January 1, 2019. We need to find the value at $t=- 2$. Substitute $t=-2$ into the function $V(t)$: [ \begin{align*} V(-2)&=14650e^{-0.168\times(-2)}\ &=14650e^{0.336}\ &\approx14650\times1.3987\ &=20481.96 \end{align*} ]

Answer:

a) -2461.20 b) 20481.96