The value of a car is falling at 10% per year so that if is the purchase price of the car in dollars, its value after t years is given by . How fast is the car depreciating after 4 years?
A) dollars per year
B) dollars per year
C) dollars per year
D) dollars per year

Question

The value of a car is falling at 10% per year so that if  is the purchase price of the car in dollars, its value after t years is given by  . How fast is the car depreciating after 4 years?
A) dollars per year 
B) dollars per year 
C) dollars per year 
D) dollars per year

Quizplus · Accepted Answer

We can use the formula for exponential decay: 
V(t) = V(0) * e^(-kt) 
where V(t) is the value of the car after t years, V(0) is the initial value (purchase price), k is the decay rate (10% or 0.1), and t is time in years. 
So, we have: 
V(t) = 11eb69ff_fe4f_2a0e_af94_ed09e9f049db_TB8190_11 
V(0) = 11eb69ff_fe4f_02fd_af94_17f75d95dfc8_TB8190_11 
k = 0.1 
t = 4

Substituting these values, we get: 
11eb69ff_fe4f_2a0e_af94_ed09e9f049db_TB8190_11 = 11eb69ff_fe4f_02fd_af94_17f75d95dfc8_TB8190_11 * e^(-0.1*4)

Simplifying, we get: 
e^(-0.4) = 11eb69ff_fe4f_2bdd_af94_5d5c5fb698f2_TB8190_11

Now, we can take the derivative of V(t) with respect to t to find the rate of change: 
dV/dt = -k * V(0) * e^(-kt)

Substituting the values we have, we get: 
dV/dt = -0.1 * 11eb69ff_fe4f_02fd_af94_17f75d95dfc8_TB8190_11 * e^(-0.1*4)

Simplifying, we get: 
dV/dt = 11eb69ff_fe4f_5120_af94_2700f082ef35_TB8190_11

Therefore, the car is depreciating at a rate of 11,696 dollars per year (rounded to the nearest dollar), which corresponds to choice B.

The Value of a Car Is Falling at 10% Per