Preview Activity 3.1.1.
Suppose that at age \(20\) you have $\(20000\) and you can choose between one of two ways to use the money: you can invest it in a mutual fund that will, on average, earn \(8\)% interest annually, or you can purchase a new automobile that will, on average, depreciate \(12\)% annually. Let’s explore how the $\(20000\) changes over time.
Let \(I(t)\) denote the value of the $\(20000\) after \(t\) years if it is invested in the mutual fund, and let \(V(t)\) denote the value of the automobile \(t\) years after it is purchased.
(a)
(b)
Note that if a quantity depreciates \(12\)% annually, after a given year, \(88\)% of the quantity remains. Compute \(V(0)\text{,}\) \(V(1)\text{,}\) \(V(2)\text{,}\) and \(V(3)\text{.}\)
(c)
Based on the patterns in your computations in (a) and (b), determine formulas for \(I(t)\) and \(V(t)\text{.}\)
(d)
Use Desmos to define \(I(t)\) and \(V(t)\text{.}\) Plot each function on the interval \(0 \le t \le 20\) and record your results on the axes inthe figure below, being sure to label the scale on the axes. What trends do you observe in the graphs? How do \(I(20)\) and \(V(20)\) compare?
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