Assume that a DVD player loses 60% of its value every year it is in a video store. Suppose the initial value of the DVD player was $80. a. What multiplier would you use to calculat... Assume that a DVD player loses 60% of its value every year it is in a video store. Suppose the initial value of the DVD player was $80. a. What multiplier would you use to calculate the DVD player’s new values? b. What is the value of the DVD player after one year? After four years? c. Write a continuous function, V(t), to model the value of a DVD player after t years. d. When does the DVD player have no value? e. Sketch a graph of this function. Be sure to scale and label the axes.
Understand the Problem
The question involves a DVD player that depreciates in value by 60% every year. It asks for the multiplier used to calculate new values, the value after certain years, a continuous function to model the value over time, and a graphical representation of this function.
Answer
a. The multiplier is $0.40$. b. After one year: $32$, after four years: $2.05$. c. The continuous function is $V(t) = 80 \times (0.40)^t$. d. The DVD player approaches $0$ but never truly reaches it.
Answer for screen readers
a. The multiplier is ( 0.40 ).
b. The value after one year is ( 32 ), and after four years is ( 2.05 ).
c. The continuous function is ( V(t) = 80 \times (0.40)^t ).
d. The DVD player never truly has zero value, but approaches zero as ( t \to \infty ).
Steps to Solve
-
Determine the Multiplier
The DVD player depreciates by 60% each year, meaning it retains 40% of its value. This is calculated as:
Multiplier = $1 - 0.60 = 0.40$ -
Calculate the Value After One Year
To find the value after one year, multiply the initial value by the multiplier:
$$ V(1) = 80 \times 0.40 = 32 $$
So, the value after one year is $32. -
Calculate the Value After Four Years
For the value after four years, use the multiplier raised to the power of 4, multiplied by the initial value:
$$ V(4) = 80 \times (0.40)^4 $$
Calculating ( (0.40)^4 ): $$ (0.40)^4 = 0.0256 $$
Thus,
$$ V(4) = 80 \times 0.0256 = 2.048 $$
Therefore, the value after four years is approximately $2.05. -
Write a Continuous Function
A continuous function ( V(t) ) to model the value of the DVD player after ( t ) years is defined as:
$$ V(t) = 80 \times (0.40)^t $$ -
Determine When the DVD Player Has No Value
The value of the DVD player approaches but never actually reaches zero as time goes to infinity. Mathematically,
$$ \lim_{t \to \infty} V(t) = 0 $$
So, the DVD player never truly has zero value, but it will approach it indefinitely. -
Sketch the Graph of the Function
To sketch the graph of ( V(t) = 80 \times (0.40)^t ):
- The x-axis represents time ( t ) in years.
- The y-axis represents the value of the DVD player.
- Plot points for ( t = 0, 1, 2, 3, 4 ) using previously calculated values. Label the axes accordingly.
a. The multiplier is ( 0.40 ).
b. The value after one year is ( 32 ), and after four years is ( 2.05 ).
c. The continuous function is ( V(t) = 80 \times (0.40)^t ).
d. The DVD player never truly has zero value, but approaches zero as ( t \to \infty ).
More Information
The depreciation model shows how rapidly the value of an electronic device can decline. This analysis can be useful for understanding the economic impact of purchasing technology items.
Tips
- Forgetting to account for the percentage decrease relative to the original value rather than the reduced value after each year.
- Confusing the depreciation multiplier with the percentage lost.
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