When a certain industrial machine is t years old, its resale value will be $V ( t ) = 4,000 e ^ { - t / 4 } + 500$ dollars. How much does the resale value change between the 3rd and 4th years?
A) Decrease by $417.95
B) Decrease by $420.05
C) Decrease by $421.07
D) Decrease by $423.91

Question

Quizplus · Accepted Answer

To find the change in resale value between the 3rd and 4th years, calculate $V(4) - V(3)$. For $V(3)$, substitute $t = 3$ into the given formula:$$V(3) = 4000e^{-3/4} + 500$$For $V(4)$, substitute $t = 4$ into the formula:$$V(4) = 4000e^{-4/4} + 500 = 4000e^{-1} + 500$$Now, calculate the difference:$$V(4) - V(3) = (4000e^{-1} + 500) - (4000e^{-3/4} + 500)$$$$= 4000e^{-1} - 4000e^{-3/4}$$$$= 4000(e^{-1} - e^{-3/4})$$$$= 4000(e^{-1} - e^{-0.75})$$Using a calculator to find the exact values:$$= 4000(0.3679 - 0.4724)$$$$= 4000(-0.1045)$$$$= -418$$The closest value to this calculation is a decrease by $417.95, which corresponds to choice A.

When a Certain Industrial Machine Is T Years Old, Its V(t)=4,000e−t/4+500V ( t ) = 4,000 e ^ { - t / 4 } + 500V(t)=4,000e−t/4+500

When a Certain Industrial Machine Is T Years Old, Its $V ( t ) = 4,000 e ^ { - t / 4 } + 500$