The value of a baseball card can be modeled by exponential decay. The value will decrease at a rate of 0.2% each yeah. The card was originally valued at $250 in the year 2000.
A- Use the exponential decay formula to write an equation to model the situation.
B- Use your equation from PART A to predict the value in 2011.
C- Sketch the graph of the equation you wrote in PART A. Find its asymptote, domain, and range.
A: you have the starting value, and the decay factor, so
v(t) = 250 * 0.98^t
where t is the years since 2000
B: just plug in t=11
C: all pure exponentials look just the same, up to scaling.
A- To model the situation of the baseball card's value decreasing at a rate of 0.2% each year, we can use the exponential decay formula:
V = P * (1 - r/100)^t
Where:
V represents the value after t years,
P is the initial value,
r is the decay rate,
t is the number of years.
In this case, the initial value is $250, the decay rate is 0.2%, or 0.002 in decimal form, and t is the number of years.
Therefore, the equation to model the situation is:
V = 250 * (1 - 0.002)^t
B- To predict the value of the baseball card in 2011, we need to substitute t = 2011 - 2000 = 11 into the equation:
V = 250 * (1 - 0.002)^11
Calculating this value will give us the predicted value of the baseball card in 2011.
C- The graph of the exponential decay equation V = 250 * (1 - 0.002)^t will have an asymptote at y = 0. This is because as time approaches infinity, the value of the card will continue to decrease, but it will never reach zero.
The domain of the graph is all real numbers, as you could have any number of years since the initial value.
The range of the graph will be all positive values of V, meaning the value of the card will only decrease from the initial value and will never become negative.
To sketch the graph, you can plot a few points by selecting different values for t, and then connect them with a smooth curve.