Question

Moore’s Law In 1965, Intel cofounder Gordon Moore initiated what has since become known as Moore’s law: The number of transistors per square inch on integrated circuits will double approximately every 18 months. In the table below, the first row lists different years and the second row lists the number of transistors (in thousands) for different years.

Ignoring the listed data and assuming that Moore’s law is correct and transistors per square inch double every 18 months, which mathematical model best describes this law: linear, quadratic, logarithmic, exponential, power? What specific function describes Moore’s law?
Which mathematical model best fits the listed sample data?
Compare the results from parts (a) and (b). Does Moore’s law appear to be working reasonably well?

Accepted Answer

Step 1: Understand Moore's Law states that the number of transistors per square inch doubles approximately every 18 months. This implies exponential growth because the quantity grows by a constant factor (doubling) over equal time intervals. Step 2: Express Moore's Law mathematically as an exponential function. Let $$N(t) be $$the number of transistors at time $$t (in $$months) since a starting point $$t=0. $$The function can be written as:

$$N(t) = N_0 	imes 2^{\frac{t}{18}}$$

where $$N_0 is $$the initial number of transistors at $$t=0. $$Step 3: Analyze the given data by plotting the number of transistors against time (years) or by transforming the data using logarithms to check if the growth is exponential. For example, take the logarithm of the transistor counts and see if it forms a linear pattern over time, which would confirm exponential growth. Step 4: Compare the actual data to the exponential model. If the data closely follows the curve predicted by $$N(t) = N_0$ 	imes 2$$^{\frac{t}$${18}}$$, then Moore's Law is working well. If the data deviates significantly, consider fitting other models such as linear, quadratic, or power functions to see which fits best. Step 5: Conclude by comparing the model from part (a) (the theoretical exponential model) with the best fit model from part (b) (based on actual data). Discuss whether the exponential model accurately describes the transistor growth over the years and if Moore's Law holds true in practice.

Verified video answer for a similar problem:

Key Concepts

Exponential Growth

Mathematical Modeling

Data Analysis and Model Fitting