Question

claudia purchases a valuable item. the equation below models the value of the item, y, after x years. select the correct statement regarding the parameter 0.84.
y = 4800 (0.84)^x
show your work here
the item depreciates at an annual rate of 84% claudia purchased the item for $16
the item appreciates at an annual rate of 16% the item appreciates at an annual rate of 84%
the item depreciates at an annual rate of 16% claudia sold the item for $16

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 (as a decimal), and \( x \) is time. For growth, it's \( y = a(1 + r)^x \), where \( r \) is the growth rate.

Step2: Compare with given equation

The given equation is \( y = 4800(0.84)^x \). Comparing with \( y = a(1 - r)^x \), we have \( 1 - r = 0.84 \).

Step3: Solve for decay rate \( r \)

Subtract \( 0.84 \) from \( 1 \): \( r = 1 - 0.84 = 0.16 \), which is \( 16\% \) as a percentage. Since \( 0.84 < 1 \), the value is decreasing (depreciating) over time.

Step4: Analyze other options

• "The item depreciates at an annual rate of 84%": Incorrect, as \( r = 0.16 = 16\% \), not 84%.
• "Claudia purchased the item for $16": Incorrect, initial value LXI0, so she purchased it for $4800.
• "The item appreciates at an annual rate of 16%": Incorrect, since \( 0.84 < 1 \), it's depreciation, not appreciation.
• "The item appreciates at an annual rate of 84%": Incorrect, it's depreciation, and rate is 16%, not 84%.
• "Claudia sold the item for $16": Incorrect, the equation models value over time, initial value is 4800, not related to selling for $16.

Answer:

The item depreciates at an annual rate of 16%