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 contin... 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. To the nearest ten dollars, what can Carlos expect the value of the computer to be 3 years after purchase?
Understand the Problem
The question is asking us to formulate an exponential equation that represents the depreciation of the value of a computer over time, using given values. We need to write it in the form y = a(b)^x, where y is the value of the computer after x years, a is the initial value, and b is the base which represents the rate of depreciation. Additionally, we want to predict the value of the computer after 3 years.
Answer
The equation is \( y = 1350(0.6)^x \), and the value after 3 years is $290.
Answer for screen readers
The exponential equation is ( y = 1350(0.6)^x ) and the predicted value after 3 years, to the nearest ten dollars, is $290.
Steps to Solve
- Identify Initial Value (a)
The initial value of the computer is the purchase price, which is $1,350. Thus, we have: $$ a = 1350 $$
- Determine Value After One Year (y)
After one year, the value of the computer is given as $810. So we can write: $$ y = 810 \text{ when } x = 1 $$
- Set Up the Exponential Equation
Using the values of (a) and (y), we can substitute into the formula ( y = a(b)^x ): $$ 810 = 1350(b)^1 $$
- Solve for Base (b)
To isolate (b), divide both sides by 1350: $$ b = \frac{810}{1350} $$
- Simplify the Value of (b)
Calculating the fraction: $$ b = \frac{810}{1350} = \frac{81}{135} = \frac{27}{45} = \frac{3}{5} = 0.6 $$
- Write the Exponential Model
Now that we have (a) and (b), we can write the equation: $$ y = 1350(0.6)^x $$
- Predict Value After 3 Years
To find the value after 3 years, substitute (x = 3) into the equation: $$ y = 1350(0.6)^3 $$
- Calculate the Value After 3 Years
Calculating ( (0.6)^3 ): $$ (0.6)^3 = 0.216 $$ Now substitute back: $$ y = 1350 \times 0.216 $$ $$ y = 291.6 $$
- Round to Nearest Ten Dollars
Rounding 291.6 to the nearest ten gives us: $$ y \approx 290 $$
The exponential equation is ( y = 1350(0.6)^x ) and the predicted value after 3 years, to the nearest ten dollars, is $290.
More Information
Carlos's computer value decreases over time. The base of 0.6 indicates a 40% depreciation annually, and these types of calculations are common in finance for predicting asset values.
Tips
- Mixing up the values of (a) and (b).
- Forgetting to round to the nearest dollar or ten dollars as instructed.
- Incorrectly calculating (b) when simplifying the fraction.
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