Question

cecily purchases a valuable item. the equation below models the value of the item, y, after x years. select the correct statement regarding the parameter 4100.

$y = 4100(1.24)^x$

show your work here

\bigcirc cecily purchased the item for $4100 \quad \bigcirc$ the item loses 41% of value each year
\bigcirc cecily sold the item for $4100 \quad \bigcirc$ the item appreciates at an annual rate of 41%
\bigcirc the item gains 41% of value each year \quad \bigcirc$ the item depreciates at an annual rate of 41%

Explanation:

Step1: Recall exponential growth formula

The general form of an exponential growth equation is \( y = a(1 + r)^x \), where \( a \) is the initial value, \( r \) is the growth rate, and \( x \) is time. For depreciation, it's \( y = a(1 - r)^x \), with \( r \) as depreciation rate.
Here, the equation is \( y = 4100(1.24)^x \). Comparing to \( y = a(1 + r)^x \), we have \( a = 4100 \) and \( 1 + r = 1.24 \), so \( r = 0.24 \) (24% growth rate).

Step2: Analyze each option

• "Cecily purchased the item for $4100": When LXI0 (initial time, year 0), \( y = 4100(1.24)^0 = 4100(1) = 4100 \). So initial value (purchase price) is $4100. This matches.
• "Cecily sold the item for $4100": Selling price would be at some \( x > 0 \), but initial value is 4100 (purchase), so this is wrong.
• "The item gains 41% of value each year": Growth rate \( r = 0.24 = 24\% \), not 41%. Wrong.
• "The item loses 41% of value each year": The base \( 1.24 > 1 \), so it's growth, not depreciation. Wrong.
• "The item appreciates at an annual rate of 41%": \( r = 24\% \), not 41%. Wrong.
• "The item depreciates at an annual rate of 41%": It's growing (base > 1), so no depreciation. Wrong.

Answer:

Cecily purchased the item for $4100