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Electric vehicles displaced 3% of automobile gasoline use, per year, at the end of 2019. This is exponential growth, since the growth depends on a percentage of the initial amount. How long will it take to displace half of the automobile gasoline use (at 3%/yr more displacement), using the exponential equation or its simplification?

Hello,

This question appears trivial (and in all likelihood it is), however It's been quite a while since I've had any sort of math education and so this question is tripping me up. I've tried plugging the 0.03 (3%) into the exponential equation y=a(1+0.03)^x, but this has yielded an answer that is unrealistically low. All other variants of the exponential equation I've tried have yielded either absurdly high or low time estimates, by orders of magnitude. Any insights into this quite simple question will be greatly appreciated.

Robert
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    The equation that I would write is simply $$ 0.03 \cdot (1.03)^x = 0.5 $$ What value do you get for $x$ ? – Matti P. Apr 15 '20 at 08:39
  • I had used 0.5 = 0.03(1.03)^x, which yielded 95.18 yrs and seemed unrealistically long. It seems that I had made an arithmetic error previously, thank you for your assistance. – Robert Apr 15 '20 at 08:42
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    Well, just look at integer values of $x$. On year $0$, you have $3~%$. A year later, this number has grown by 3 per cent to a "whopping" $3.09%$. A year from that, it's $3.183%$. As you see, the beginning is indeed quite slow .. The question arises: When we say that the figure increases by 3 per cent, do we mean actually 3 per cent or 3 percentage points? – Matti P. Apr 15 '20 at 08:45
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    If we calculate with 3 percentage point increase per year, we have the following logic: On year 0, we have 3%. On year 1, we have 6% etc ... So this is really much faster. – Matti P. Apr 15 '20 at 08:46
  • In this situation I believe the question was indeed referring to the figure increasing by 3 percent rather 3 percentage points. Thank you for answering my question, how can I mark it as answered? I don't see an option to do that for comments (or should I delete this question since it's so trivial?) – Robert Apr 15 '20 at 08:47

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If we calculate with an actual $3$ percent per year increase (instead of percentage points), the model that we can use to calculate the figure on year $x$ is simply $$ 0.03 \cdot (1.03)^x $$ According to this model, one should wait over $95$ years in order for this to increase above $0.5$.

Matti P.
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