Problem 75 - Precalculus : Building Concepts and Connections

Question

In \(1965,\) Gordon Moore, then director of Intel research, conjectured that the number of transistors that fit on a computer chip doubles every few years. This has come to be known as Moore's Law. Analysis of data from Intel Corporation yields the following model of the number of transistors per chip over time: $$s(t)=2297.1 e^{0.3316 t}$$ where \(s(t)\) is the number of transistors per chip and \(t\) is the number of years since \(1971 .\) (Source: Intel Corporation) (a) According to this model, what was the number of transistors per chip in \(1971 ?\) (b) How long did it take for the number of transistors to double?

Step-by-Step Solution

Answer

According to this model, in 1971 there were approximately 2297.1 transistors per chip and it took roughly 2.09 years for the number of transistors to double.

Step 1: Determine Transistors in 1971

Substitute \(t = 0\) into the given model \(s(t) = 2297.1 e^{0.3316 t}\), to find the number of transistors per chip in 1971.

Step 2: Calculation for 1971

This gives \(s(0) = 2297.1 * e^{0.3316 * 0} = 2297.1 * e^0 = 2297.1\). So in 1971 there were approximately 2297.1 transistors per chip.

Step 3: Determine Time for Transistor Count to Double

To find out how long it takes for the number of transistors to double, set the model equal to twice the transistor count in 1971: \(2 * 2297.1 = 4594.2 = 2297.1 * e^{0.3316 t}\).

Step 4: Solve for t

To isolate \(t\), first divide both sides by 2297.1: \(2 = e^{0.3316 t}\). Then take the natural logarithm of both sides (since the base of the exponential function is \(e\)), ln(2) = 0.3316*t. Finally, divide by 0.3316 to find \(t = ln(2) / 0.3316\). After the calculation, we get approximately \(t = 2.09\) years. So, it took roughly 2.09 years for the number of transistors to double according to the model.

Key Concepts

Transistors

Transistors play an essential role in modern electronics. They are tiny semiconductor devices that function as the building blocks of most electronic circuits, including those on a computer chip. As a switch or amplifier, a transistor can control the flow of electrical current in a circuit.
Here are simple facts about transistors:

• Transistors are what allow computers and other electronic devices to perform calculations and store information.
• With the ability to control electrical signals, transistors are crucial for creating integrated circuits, also known as "computer chips" or "microchips."
• A higher number of transistors on a chip generally means more computing power and efficiency.

In the context of computational advancements, the count of transistors on a single chip over time can be used to measure technological progress. This increase is not just a mark of quantity but also a sign of enhanced capability.

Moore's Law

Moore's Law is named after Gordon Moore, one of the co-founders of Intel. It's an observation made in 1965 predicting that the number of transistors on a computer chip would approximately double every two years.
Here is why Moore's Law is significant:

• This law implies not just a doubling of transistors, but also indicates exponential improvements in processing speed and energy efficiency at a constant cost over time.
• It has driven the semiconductor industry forward by inspiring predictions and future goals for increasing chip performance.
• Though not a physical law, it has been remarkably accurate for many decades, guiding research and development.

However, as transistors reach atomic scales, sustaining this growth according to Moore's Law becomes more challenging. As of now, the prediction is reaching its limits with new technologies being explored to continue progress.

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to the power of a variable exponent. In our context, it is used to model the growth of transistors over time.
In the formula provided: \[s(t) = 2297.1 e^{0.3316 t}\]The exponential nature of this function means:

• As time (\(t\)) increases, the value of \(s(t)\), the number of transistors, grows rapidly.
• The function implies sustained and compounding growth, a hallmark characteristic of exponential changes.
• Exponential functions excel at modeling real-world situations where growth accelerates as time progresses, like technology development.

Understanding exponential growth can elucidate how technological advancements happen at a seemingly ever-increasing pace. In Moore's model, it helps explain how relatively short periods can lead to substantial increases in transistor counts.