Question

the price p, in dollars, of a specific car that is x years old is modeled by the following function. p(x)=45,285·0.90^x (a) how much should a 3 - year - old car cost? (b) how much should a 6 - year - old car cost? (c) explain the meaning of the base 0.90 in this problem. (a) a 3 - year - old car should cost approximately $□. (round to the nearest whole number as needed.)

Explanation:

Step1: Substitute x = 3 into the function

$p(3)=45285\times0.90^{3}$

Step2: Calculate $0.90^{3}$

$0.90^{3}=0.90\times0.90\times0.90 = 0.729$

Step3: Calculate the price

$p(3)=45285\times0.729 = 32912.765\approx32913$

Step4: Substitute x = 6 into the function for part (b)

$p(6)=45285\times0.90^{6}$

Step5: Calculate $0.90^{6}$

$0.90^{6}=0.90\times0.90\times0.90\times0.90\times0.90\times0.90=0.531441$

Step6: Calculate the price for 6 - year - old car

$p(6)=45285\times0.531441\approx24064$

Step7: Explain the base 0.90 for part (c)

The base 0.90 represents the decay factor. Each year, the value of the car is 90% (or 0.90 times) of its value the previous year.

Answer:

(a) 32913
(b) 24064
(c) Each year, the value of the car is 90% of its value the previous year.