Solve the problem.
-The value V of a car that is t years old can be modeled by V(t)= 19,518(0.82)t. According to the model, when will the car be worth $6000? Round to the nearest tenth of a year.
A)0.9 yr
B)51.1 yr
C)5.9 yr
D)1.9 yr

Question

Quizplus · Accepted Answer

We are given the function V(t)= 19,518(0.82)t, where V(t) represents the value of the car that is t years old.

We need to find out when the car will be worth $6000.

So we set up the equation:

6000 = 19,518(0.82)t

Divide both sides by 19,518:

0.3074 = (0.82)t

Take the logarithm of both sides:

log(0.3074) = log(0.82)t

Use the power rule of logarithms:

log(0.3074) = t(log0.82)

Divide both sides by log0.82:

t = log(0.3074) / log(0.82)

Using a calculator, we get:

t ≈ 5.9

Therefore, the car will be worth $6000 when it is approximately 5.9 years old.

The answer is (C) 5.9 yr. We need to solve the equation V(t) = 6000 for t. 
6000 = 19518(0.82)^t 
0.82^t = 0.307 
Taking the logarithm of both sides: 
tlog(0.82) = log(0.307) 
t = log(0.307) / log(0.82) 
t ≈ 5.9 years 
Therefore, the car will be worth $6000 in approximately 5.9 years, which is answer choice C.

Solve the Problem. -The Value V of a Car That Is T Years