Question

roger purchases a valuable item. the equation below models the value of the item, y, after x years. select the correct statement regarding the parameter 0.99.
y = 8100 (0.99)^x
show your work here

○ roger purchased the item for $1 ○ the item depreciates at an annual rate of 1%.
○ the item depreciates at an annual rate of 99%. ○ the item appreciates at an annual rate of 99%.
○ roger sold the item for $1 ○ the item appreciates at an annual rate of 1%

Explanation:

Step1: Recall exponential decay formula

The general form of an exponential decay function is \( y = a(1 - r)^x \), where \( a \) is the initial amount, \( r \) is the rate of decay (depreciation) per period, and \( x \) is the number of periods.

Step2: Compare with given equation

The given equation is \( y = 8100(0.99)^x \). We can rewrite \( 0.99 \) as \( 1 - 0.01 \). So, comparing with \( y = a(1 - r)^x \), we have \( 1 - r = 0.99 \), which means \( r = 1 - 0.99 = 0.01 \) or \( 1\% \). This indicates the item depreciates (since the base \( 0.99 < 1 \)) at an annual rate of \( 1\% \).

Step3: Eliminate other options

• "Roger purchased the item for $1" is wrong because the initial value LXI0, so the purchase price is $8100.
• "The item depreciates at an annual rate of 99%" is wrong because \( r = 0.01 = 1\% \), not 99%.
• "The item appreciates..." is wrong because the base \( 0.99 < 1 \), so it's depreciation, not appreciation.
• "Roger sold the item for $1" is wrong as the equation models value over time, not a sale price of $1.

Answer:

The item depreciates at an annual rate of 1%