Question

joanne purchases a valuable item. the equation below models the value of the item, y, after x years. select the correct statement regarding the parameter 0.93.
$y = 4800 (0.93)^x$
show your work here

the item appreciates at an annual rate of 7%
joanne purchased the item for $7
the item depreciates at an annual rate of 7%
the item appreciates at an annual rate of 93%
joanne sold the item for $7
the item depreciates at an annual rate of 93%

Explanation:

Step1: Recall exponential decay formula

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

Step2: Analyze the given equation

The given equation is \( y = 4800(0.93)^x \). Comparing with the decay formula \( y = a(1 - r)^x \), we have \( 1 - r = 0.93 \).

Step3: Solve for the rate \( r \)

Subtract 0.93 from 1: \( r = 1 - 0.93 = 0.07 \), which is 7% when converted to a percentage. Since \( 0.93 < 1 \), the value of the item is decreasing over time, so it's depreciation.

Step4: Evaluate the options

• "The item appreciates at an annual rate of 7%": Incorrect, because \( 0.93 < 1 \), it's depreciation, not appreciation.
• "Joanne purchased the item for $7": Incorrect, the initial value LXI0, so she purchased it for $4800.
• "The item depreciates at an annual rate of 7%": Correct, as we found \( r = 0.07 = 7\% \) and the function is a decay (depreciation) function.
• "The item appreciates at an annual rate of 93%": Incorrect, \( 0.93 \) is the decay factor, not a growth factor for 93% appreciation.
• "Joanne sold the item for $7": Incorrect, the equation models the value after LXI0 years, and the initial value is $4800.
• "The item depreciates at an annual rate of 93%": Incorrect, \( r = 0.07 = 7\% \), not 93%.

Answer:

The item depreciates at an annual rate of 7%