Question
Carlos purchased a new computer for $1,350. One year later, a popular tech website valued the same computer at $810. The website predicts that the value of the computer will continue depreciating each year.
Write an exponential equation in the form y=a(b)x that can model the value of the computer, y, x years after purchase.
Use whole numbers, decimals, or simplified fractions for the values of a and b.
y =
To the nearest ten dollars, what can Carlos expect the value of the computer to be 3 years after purchase?
Write an exponential equation in the form y=a(b)x that can model the value of the computer, y, x years after purchase.
Use whole numbers, decimals, or simplified fractions for the values of a and b.
y =
To the nearest ten dollars, what can Carlos expect the value of the computer to be 3 years after purchase?
Answers
GPT-4o mini
To create an exponential equation that models the value of the computer over time, we will use the given information about Carlos's computer.
1. **Initial Value (a)**: The initial value of the computer is $1,350 when it was purchased. Thus, \( a = 1350 \).
2. **Value After One Year**: The value of the computer after one year is $810. From this, we can find the depreciation factor \( b \).
To set up the equation, we can use the general form of the exponential decay formula:
\[
y = a(b)^x
\]
Substituting what we have:
\[
810 = 1350(b)^1
\]
To find \( b \), we’ll isolate it:
\[
b = \frac{810}{1350}
\]
Calculating that gives:
\[
b = \frac{81}{135} = \frac{27}{45} = \frac{3}{5} = 0.6
\]
Now, we have our values:
- \( a = 1350 \)
- \( b = 0.6 \)
Therefore, the equation that models the value of the computer \( y \) after \( x \) years is:
\[
y = 1350(0.6)^x
\]
Next, we want to calculate the value of the computer after 3 years:
\[
y = 1350(0.6)^3
\]
Calculating \( (0.6)^3 \):
\[
(0.6)^3 = 0.216
\]
Now substituting this back into the equation:
\[
y = 1350 \times 0.216 = 291.6
\]
To the nearest ten dollars, the expected value of the computer 3 years after purchase is:
\[
\text{Value} \approx 290 \, \text{dollars}
\]
Final answers:
1. The equation is \( y = 1350(0.6)^x \).
2. The value of the computer 3 years after purchase is approximately $290.
1. **Initial Value (a)**: The initial value of the computer is $1,350 when it was purchased. Thus, \( a = 1350 \).
2. **Value After One Year**: The value of the computer after one year is $810. From this, we can find the depreciation factor \( b \).
To set up the equation, we can use the general form of the exponential decay formula:
\[
y = a(b)^x
\]
Substituting what we have:
\[
810 = 1350(b)^1
\]
To find \( b \), we’ll isolate it:
\[
b = \frac{810}{1350}
\]
Calculating that gives:
\[
b = \frac{81}{135} = \frac{27}{45} = \frac{3}{5} = 0.6
\]
Now, we have our values:
- \( a = 1350 \)
- \( b = 0.6 \)
Therefore, the equation that models the value of the computer \( y \) after \( x \) years is:
\[
y = 1350(0.6)^x
\]
Next, we want to calculate the value of the computer after 3 years:
\[
y = 1350(0.6)^3
\]
Calculating \( (0.6)^3 \):
\[
(0.6)^3 = 0.216
\]
Now substituting this back into the equation:
\[
y = 1350 \times 0.216 = 291.6
\]
To the nearest ten dollars, the expected value of the computer 3 years after purchase is:
\[
\text{Value} \approx 290 \, \text{dollars}
\]
Final answers:
1. The equation is \( y = 1350(0.6)^x \).
2. The value of the computer 3 years after purchase is approximately $290.