Respuesta :
Answer:
D) 9 years.
Step-by-step explanation:
We have been given that Gary bought a car for $40,000 and equation [tex]V=40,000(0.85)^t[/tex] represents the value of the car after t years.
First of all we will find the one-fourth of 40,000.
[tex]\text{One-forth of car's purchase price}=\frac{\$40,000}{4}[/tex]
[tex]\text{One-forth of car's purchase price}=\$10,000[/tex]
To find the time it will take the car to be worth less than one-fourth of its purchase price, we will substitute V=10,000 in our given equation.
[tex]10,000=40,000(0.85)^t[/tex]
Let us divide both sides of our equation by 40,000.
[tex]\frac{10,000}{40,000}=\frac{40,000(0.85)^t}{40,000}[/tex]
[tex]0.25=0.85^t[/tex]
Let us take natural log of both sides of our equation.
[tex]ln(0.25)=ln(0.85^t)[/tex]
Using natural log property [tex]ln(a^b)=b*ln(a)[/tex] we will get,
[tex]ln(0.25)=t*ln(0.85)[/tex]
[tex]\frac{ln(0.25)}{ln(0.85)}=\frac{t*ln(0.85)}{ln(0.85)}[/tex]
[tex]\frac{-1.3862943611198906}{-0.1625189294977749}=t[/tex]
[tex]8.530048563597=t[/tex]
Upon rounding our answer to the nearest year we will get,
[tex]t\approx 9[/tex]
Therefore, it will take 9 years the car to be worth less than one-fourth of its purchase price and option D is the correct choice.
