39–40. LED lighting LED (light-emitting diode) bulbs are rapid...

Question

39–40. LED lighting LED (light-emitting diode) bulbs are rapidly decreasing in cost, and they are more energy-efficient than standard incandescent light bulbs and CFL (compact fluorescent light) bulbs. By some estimates, LED bulbs last more than 40 times longer than incandescent bulbs and more than 8 times longer than CFL bulbs. Haitz’s law, which is explored in the following two exercises, predicts that over time, LED bulbs will exponentially increase in efficiency and exponentially decrease in cost.

Haitz’s law predicts that the cost per lumen of an LED bulb decreases by a factor of 10 every 10 years. This means that 10 years from now, the cost of an LED bulb will be 1/10 of its current cost. Predict the cost of a particular LED bulb in 2021 if it costs 4 dollars in 2018.

Haitz’s law predicts that the cost per lumen of an LED bulb decreases by a factor of 10 every 10 years. This means that 10 years from now, the cost of an LED bulb will be 1/10 of its current cost. Predict the cost of a particular LED bulb in 2021 if it costs 4 dollars in 2018.

Accepted Answer

Welcome back everyone. In this problem, Parker's law states that the price of a certain portable power bank falls by a factor of $$10 every 10 years. A particular power bank cost 8 in 2015. $$Using Parker's law, what will be the predicted price of the power bank in 2020? Roer answer to two decimal places. A says $$2.53. B 2.28. C$$ $$3.82 and D 1.82. $$Now, our problem tells us that Parker's law states that the price of our power bank falls by a factor of 10 every 10 years. So in other words, by Parker's law. OK. It's really telling us that our price 10 years later, T + 10, is going to be equal to our current price P of T divided by 10. Now, if we think about it here, we can also express the factor using the exponential decay formula. And by the exponential decay model, OK, by the exponential decay model. OK. Then P of T or current price would be equal to P0 or initial price multiplied by E to the negative kT where ks are constant and T is our time in years, while P of T + 10 or 10 years later would be equal to P0 multiplied by E negative K multiplied by T + 10. And if we expand that term for our power, it will be P not E to the negative KT multiplied by E to the negative 10K. No, since both of these are equal to P T plus 10, OK? P of T divided by 10 and P not. P note to the negative kt multiplied by E to the negative 10k. This must mean they're both equal to each other. So in other words, we can say that P0 E to the negative KT multiplied by Ene 10k is going to be equal to P T P E to the negative K T divided by 10. Now if we look at both sides of our equation, we can cancel P not E to the negative KT and that will leave us with E to the negative 10 divided to the negative 10K equal to 1/10. Now if we do some transposition here and take the natural log of both sides, then we'll have negative 10K on the left equal to the natural log of 10th on the right or negative 10. So this must mean that K equals negative LN10 divide by 10 or 1/10 of LN10. Now that we have a value fork, we can use it to help us solve for P because no this would this must mean then that our price P would be equal to P0 E to the negative LN 10 divided by 10 T. And since Based on what we know about logs that E to the LNA would be equal to A, OK, then P should be equal to P multiplied by E to the LN10 multiplied by or raised to the power of negative T divided by 10. And by our identity this must simplified then to P not multiplied by 10 of the negative of T. Since we have an expression known for a price, then we can solve. Because remember earlier we know that or we were told that the power bank cost $$8 in 2015 and since we're starting with 2015 when T equals 0 OK$$, that is zero years from 2015, then PO must be equal to $$8. PO is our original price. No, we can use that in our formula to solve in 2020. Now in 2020, that's 5 years later, so T equals 5, which means that P will be equal to 8 multiplied by 10 to 50, which rounded to two decimal places would be approximately 2.53. $$Thus our poor bank will, or it's predicted that our poor bank will cost $2.53 in 2020. If we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Exponential Decay

Haitz’s Law

Time Interval and Proportionality in Exponential Models

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