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.

Accepted Answer

Identify the type of change described: The problem states that the cost decreases by a factor of 10 every 10 years, which indicates an exponential decay model. Set up the exponential decay formula for cost: Let $$C(t)$$ represent the cost at time $$t$$ years, and $$C_0 be $$the initial cost. The formula is $$C(t) = C_0$ 	imes a$$^{\frac{t}$${T}}$$, where $$a is $$the decay factor and $$T is $$the time period for one decay step. Assign the known values: The initial cost $$C_0 is 4 $$dollars in 2018, the decay factor $$a is$$ $$\frac{1}{10} ($$since cost decreases by a factor of 10), and the time period $$T is 10 $$years. Calculate the time difference $$t$$ between 2018 and 2021: $$t = 2021 - 2018 = 3$$ years. Substitute all values into the formula to express the cost in 2021: $$C(3) = 4 	imes \left(\frac{1}{10}\$$right)^{\frac{3}$${10}}. $$This expression represents the predicted cost in 2021.

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Concetti chiave

Exponential Decay

Haitz’s Law

Time Interval and Proportionality in Exponential Models