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 dollars per year (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 dollars per year (type an integer or decimal rounded to two decimal places as needed )

Answer

Explanation:

Step1: Find the derivative of $V(t)$

The derivative of $V(t)=14650e^{- 0.168t}$ using the chain - rule. If $y = ae^{bx}$ where $a = 14650$ and $b=-0.168$, then $V^\prime(t)=a\times b\times e^{bx}$. So $V^\prime(t)=14650\times(-0.168)e^{-0.168t}=-2461.2e^{-0.168t}$.

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

Substitute $t = 0$ into $V^\prime(t)$. We know that $e^{0}=1$. So $V^\prime(0)=-2461.2\times e^{0}=-2461.20$.

Step3: Find the original value of the equipment

To find the original value of the equipment on January 1, 2017 (2 years before January 1, 2019, so we need to consider $t=- 2$ in the reverse - time sense, but we can also think of it in terms of the general formula. The value of the equipment at $t = 0$ (January 1, 2019) is $V(0)=14650$. The general formula for exponential decay is $V(t)=V_0e^{-kt}$, where $V_0$ is the initial value, $k$ is the decay constant. When $t = 0$, $V(0)=V_0e^{0}=V_0$. So the original value $V_0$ can be found by considering the fact that the value at $t = 0$ is $14650$ and working backward. If we assume the decay formula holds from the start of 2017, we can also note that when $t = 0$ (January 1, 2019), $V(0)=14650$. The original value of the equipment on January 1, 2017, we can use the fact that the value at $t = 0$ is $V(0)$ and the decay formula $V(t)=V_0e^{-kt}$. When $t = 0$, $V(0)=V_0$. In another way, if we consider the value at $t = 0$ as $14650$ and know the decay formula $V(t)=V_0e^{-kt}$, we can find that when $t = 0$, $V_0 = 14650\div e^{0}=14650$. But if we use the idea of going back in time, we know that the value at $t = 0$ is $V(0)$ and the formula $V(t)=V_0e^{-kt}$. Let's assume the decay starts from the initial purchase. We know that $V(t)=V_0e^{-kt}$, when $t = 2$ (from 2017 to 2019), $V(2)=14650$, so $14650=V_0e^{-0.168\times2}$. Then $V_0=\frac{14650}{e^{- 0.336}}=14650e^{0.336}\approx14650\times1.3987=20490.96$.

Answer:

a) -2461.20 b) 20490.96