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The price [tex]\( p \)[/tex], in dollars, of a specific car that is [tex]\( x \)[/tex] years old is modeled by the function below.

[tex]\[ p(x) = 22,285(0.92)^x \][/tex]

(a) How much should a 4-year-old car cost?

(b) How much should a 7-year-old car cost? (Round to the nearest whole number as needed.)

A 7-year-old car should cost approximately [tex]\( \)[/tex] [tex]\(\$\square\)[/tex] (Round to the nearest whole number as needed.)

Sagot :

GinnyAnsweridn
Sure, let's solve these problems step by step.

We have the function:
[tex]\[ p(x) = 22,285 \times (0.92)^x \][/tex]

### (a) Cost of a 4-year-old car
To find the cost of a 4-year-old car, we will substitute [tex]\( x = 4 \)[/tex] into the function:

1. Substitute [tex]\( x = 4 \)[/tex] into the function:
[tex]\[ p(4) = 22,285 \times (0.92)^4 \][/tex]

2. Calculate [tex]\( (0.92)^4 \)[/tex]:
[tex]\[ (0.92)^4 \approx 0.71639 \][/tex]

3. Multiply this result by 22,285:
[tex]\[ 22,285 \times 0.71639 \approx 15,965 \][/tex]

So, the cost of a 4-year-old car is approximately [tex]\(\$15,965\)[/tex].

### (b) Cost of a 7-year-old car
To find the cost of a 7-year-old car, we will substitute [tex]\( x = 7 \)[/tex] into the function:

1. Substitute [tex]\( x = 7 \)[/tex] into the function:
[tex]\[ p(7) = 22,285 \times (0.92)^7 \][/tex]

2. Calculate [tex]\( (0.92)^7 \)[/tex]:
[tex]\[ (0.92)^7 \approx 0.56952 \][/tex]

3. Multiply this result by 22,285:
[tex]\[ 22,285 \times 0.56952 \approx 12,432 \][/tex]

So, the cost of a 7-year-old car is approximately [tex]\(\$12,432\)[/tex].

In summary:
- A 4-year-old car should cost approximately [tex]\(\$15,965\)[/tex].
- A 7-year-old car should cost approximately [tex]\(\$12,432\)[/tex].
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