# Thread: help with an exponential function

1. ## help with an exponential function

Hi , I need some help with the next problem

In 1965, Gordon Moose observed that the amount of computing power possible to put on a chip doubles every two years. In 1990, there were 1,000,000 transistors per chip. How many transistors per chip were there in 2000? 1985?

I have tried to solve it ; however, I don't know how to interpret its growth ( I mean when it doubles every 2 years). Can some help me to write a function that describes this problem? Thanks in advance!!

2. ## Exponential growth

Hello skorpiox
Originally Posted by skorpiox
Hi , I need some help with the next problem

In 1965, Gordon Moose observed that the amount of computing power possible to put on a chip doubles every two years. In 1990, there were 1,000,000 transistors per chip. How many transistors per chip were there in 2000? 1985?

I have tried to solve it ; however, I don't know how to interpret its growth ( I mean when it doubles every 2 years). Can some help me to write a function that describes this problem? Thanks in advance!!
We need a function that gives $\displaystyle n$, the number of transistors after a time t years (measured from 1990), such that

At time $\displaystyle t = 0, n = 10^6$

And, since the number doubles every 2 years, at time $\displaystyle t = 2, n = 10^6 \times 2$

At time $\displaystyle t = 4, n = 10^6 \times 2^2$

...and so on.

Clearly, we are going to need something like:

$\displaystyle n = 10^6 \times 2^{at + b}$ for some constants $\displaystyle a$ and $\displaystyle b$.

Notice that at time $\displaystyle t = 0, n = 10^6 \times 1 = 10^6 \times 2^0$

and at time $\displaystyle t = 2, n = 10^6 \times 2^1$

So we want $\displaystyle t = 0$ to give $\displaystyle at + b = 0$

$\displaystyle \Rightarrow b = 0$

and $\displaystyle t = 2$ to give $\displaystyle at + b = 1$

$\displaystyle \Rightarrow 2a + 0 = 1$

$\displaystyle \Rightarrow a = 0.5$

So the function we need is

$\displaystyle n = 10^6 \times 2^{0.5t}$

You now need to find the value of $\displaystyle n$ when $\displaystyle t = 10$, and when $\displaystyle t = -5$.

Can you complete it now?