<p>The value of an antique car is modeled by the function V(x)=107,000(1.009)^(2x/3) where x is the number of years since 2005. By what approximate percent rate is the value of the car increasing per year?</p>
<p>How could one calculate this answer with a calculator? I am not sure how to raise 1.009 to the 2/3 power using my basic calculator. Thanks.</p>
<p>You square 1.009, then cube root it, since a number to the (1/x) power is the xth root of that number. You could rewrite 1.009(2/3) as 1.009^(2^(1/3)).</p>
<p>I think this is possible to answer without even writing anything. I read somewhere that the formula for these types of questions (namely, exponential growth) is: x(1+ growthrate%)^(period of time)
Plugging in the info provided:
x = 107,000
Growth Rate = .009 (0.9%)
Though I could be totally wrong here. If that weren’t one of the options, then I’d substitute x for 1 and then 2 and see how much percent it got increased by. </p>
<p>If you look at the binominal expansion of (1+a)^n, if a is much smaller than 1, the first two terms give a pretty good approximation, ie. 1 + na because a^2 becomes much smaller as long as |a| << 1.
Look at what happens after 1 year when you subtract the value at the end of the first year from the base value, and divide the result by the base value.
The 107,000 is a multiplier in both and cancels out. So you have (1 + 2/3 (0.009) ) - 1 and you’re left with an approximate value of 0.006 as the fraction that the value has increased.</p>