Exponential Growth and Decay Question 10

calculator allowed Mike's computer can perform 1 million calculations in one second. Moore's Law says that computer performance doubles every 2 years. Mike always upgrades his computer to the latest technology, so in 4 years, Mike's computer will do 4 million calculations per second according to Moore's Law.

To run a popular video game Crysis, Mike needs a computer that can do 3.84 billion calculations per minute. In how many years will Mike's computer be able to run the video game?

2

4

6

8

Approach

We should first convert the given rate 3.84 billion per minute to match the units given in the initial description of Moore's Law.

\frac{3.84 \text{ billion calculations}}{1 \text{ minute}} \cdot \frac{1{,}000 \text{ million calculations}}{1 \text{ billion calculations}} \cdot \frac{1\text{ minute}}{60\text{ seconds}}

= 64 \text{ million calculations per second}

We can construct a simple exponential model or use a table. Below I demonstrate both options:

Exponential Model (initial value=1, rate=2, final value=64)

y=ab^x

64=1(2)^x

64=2^x

2^6=2^x

x=\boxed{6 \text{ years}}

Table

\begin{array}{|c|c|}\hline \text{Year} & \text{Million calculations} \\ & \text{per second} \\ \hline 0 & 1 \\ \hline 1 & 2 \\ \hline 2 & 4 \\ \hline 3 & 8 \\ \hline 4 & 16 \\ \hline 5 & 32 \\ \hline \colorbox{aqua}{$6$} & \colorbox{aqua}{$64$} \\ \hline \end{array}