1. According to Moore’s law, the number of transistors T on a semiconductor chip doubles every two years. Assuming a continuous growth in transistors over time, what is the annual growth rate ( in %) of transistors, implied by Moore’s law?
To determine the annual growth rate implied by Moore's Law, we can use the compound interest formula:
A = P * (1 + r)^n
where:
A = final value
P = initial value
r = growth rate (as a decimal)
n = number of periods
In the case of Moore's Law, we assume that the number of transistors doubles every two years. Therefore, P would be the initial number of transistors, and A would be the final number of transistors after one year.
Since the number of transistors doubles every two years, we can express this growth rate as an equation: P * (1 + r)^2 = A
To solve for r, we need to isolate it in the equation. First, divide both sides of the equation by P, then take the square root of both sides:
(1 + r)^2 = A / P
1 + r = sqrt(A / P)
Now, subtract 1 from both sides to get the growth rate:
r = sqrt(A / P) - 1
In our case, we want the growth rate per year, so we need to divide the two-year growth rate by 2:
r = [sqrt(A / P) - 1] / 2
To determine the growth rate as a percentage, we would multiply the result by 100. So the formula to calculate the annual growth rate (in %) implied by Moore's Law would be:
Annual growth rate = 100 * [sqrt(A / P) - 1] / 2
By plugging in the appropriate values for A (final number of transistors) and P (initial number of transistors), you can calculate the annual growth rate implied by Moore's Law.