Final answer:
After running the calculations, none of the options provided (A, B, C, or D) are correct. The collectible, depreciating for three years and then appreciating, will still be worth less than $100 after 7 years.
Step-by-step explanation:
The question is asking us to determine how long it will take for a collectible, bought at $80 and subject to depreciation and appreciation over time, to exceed the value of $100. The collectible loses 5% of its value each year for the first three years and then increases in value by 3% each year thereafter.
Let's calculate the value at the end of each year up to the point where the collectible's value exceeds $100.
- End of Year 1: $80 - (5% of $80) = $80 - $4 = $76
- End of Year 2: $76 - (5% of $76) = $76 - $3.80 = $72.20
- End of Year 3: $72.20 - (5% of $72.20) = $72.20 - $3.61 = $68.59
Starting from the end of Year 3, the collectible's value starts to increase by 3% annually. We'll keep calculating the value at the end of each year until it exceeds $100.
- End of Year 4: $68.59 + (3% of $68.59) = $68.59 + $2.06 = $70.65
- End of Year 5: $70.65 + (3% of $70.65) = $70.65 + $2.12 = $72.77
- End of Year 6: $72.77 + (3% of $72.77) = $72.77 + $2.18 = $74.95
- End of Year 7: $74.95 + (3% of $74.95) = $74.95 + $2.25 = $77.20
We can see from this pattern that it will take more than seven years for the collectible to exceed a value of $100. Therefore, none of the given options A, B, C, or D are correct, as it will take longer than 7 years for the collectible to be worth more than $100.