Moore's Law predicts that the number of transistors that can be fit on a microchip will increase by 40% every year. If microchips from a given year could hold about 922,200 transis... Moore's Law predicts that the number of transistors that can be fit on a microchip will increase by 40% every year. If microchips from a given year could hold about 922,200 transistors, how many transistors could fit on a microchip 19 years later? If necessary, round your answer to the nearest whole number.
Understand the Problem
The question is asking us to calculate the number of transistors that can fit on a microchip based on Moore's Law, which states an increase of 40% each year. We need to apply this growth rate over 19 years to the initial amount of 922,200 transistors.
Answer
$10,395,594,568$
Answer for screen readers
The total number of transistors that can fit on the microchip after 19 years is approximately $10,395,594,568$.
Steps to Solve
- Identify the Growth Formula
To calculate the number of transistors after a certain number of years with a growth rate, we can use the formula for exponential growth:
$$ N = N_0 (1 + r)^t $$
where:
- $N_0$ is the initial number of transistors (922,200),
- $r$ is the growth rate (40% or 0.4),
- $t$ is the number of years (19).
- Substitute Values into the Formula
Now we will substitute the values into the formula:
$$ N = 922,200 \times (1 + 0.4)^{19} $$
This simplifies to:
$$ N = 922,200 \times (1.4)^{19} $$
- Calculate the Growth Factor
Next, we need to compute $(1.4)^{19}$. This can be calculated using a calculator or software:
$$ (1.4)^{19} \approx 11251.5747 $$
- Calculate the Final Number of Transistors
Now we multiply the initial number of transistors by the calculated growth factor:
$$ N \approx 922,200 \times 11251.5747 $$
- Complete the Calculation
Finally, calculate the total:
$$ N \approx 10,395,594,568 $$
This result means that approximately 10,395,594,568 transistors can fit on the microchip after 19 years.
The total number of transistors that can fit on the microchip after 19 years is approximately $10,395,594,568$.
More Information
This result illustrates how exponential growth, as described by Moore's Law, can lead to vast increases in the number of transistors over time. The computing power available has dramatically improved due to the advancements in technology.
Tips
- Confusing growth rates: Ensure to convert the percentage growth rate into a decimal form correctly (e.g., 40% = 0.4).
- Incorrectly applying the formula: Always ensure you are using the correct formula for exponential growth and not a linear growth formula.
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