The lifetime t (in weeks) of a light bulb can be defined by the force of mortality 0.25t2 +0.4t.A hotel has five hundred rooms, and has just opened with four new light bulbs in eachroom. Light bulbs are checked exactly once a week and all defective bulbs are replacedimmediately.(a) Calculate the number of bulbs that the hotel would expect to replace at the end ofeach of the first, second, and third week of opening
Question
The lifetime t (in weeks) of a light bulb can be defined by the force of mortality 0.25t2 +0.4t.A hotel has five hundred rooms, and has just opened with four new light bulbs in eachroom. Light bulbs are checked exactly once a week and all defective bulbs are replacedimmediately.(a) Calculate the number of bulbs that the hotel would expect to replace at the end ofeach of the first, second, and third week of opening
Solution
To solve this problem, we need to calculate the probability of a bulb failing in the first, second, and third week. The force of mortality function given is 0.25t^2 + 0.4t. This function gives the instantaneous rate of failure at time t.
(a) For the first week (t=1), the force of mortality is 0.25(1)^2 + 0.4(1) = 0.65. This is the probability of a bulb failing in the first week.
The hotel has 500 rooms with 4 bulbs each, so there are 500*4 = 2000 bulbs.
The expected number of bulbs to fail in the first week is therefore 2000 * 0.65 = 1300 bulbs.
(b) For the second week (t=2), the force of mortality is 0.25(2)^2 + 0.4(2) = 1.8. This is the probability of a bulb failing in the second week.
The expected number of bulbs to fail in the second week is therefore 2000 * 1.8 = 3600 bulbs.
(c) For the third week (t=3), the force of mortality is 0.25(3)^2 + 0.4(3) = 3.05. This is the probability of a bulb failing in the third week.
The expected number of bulbs to fail in the third week is therefore 2000 * 3.05 = 6100 bulbs.
So, the hotel would expect to replace 1300 bulbs at the end of the first week, 3600 bulbs at the end of the second week, and 6100 bulbs at the end of the third week.
Please note that these calculations assume that the force of mortality is the same for all bulbs, which may not be the case in reality. Also, the calculations do not take into account the fact that bulbs that fail are replaced, which would reduce the number of bulbs that can fail in subsequent weeks.
Similar Questions
The lifespan of a particular brand of CFL light bulb is uniformly distributed between 19 and 39 months. Assume lifespans are independent from light bulb to light bulb. In a package that contains two of these CFL light bulbs, find the probability that the first bulb lasts for more than 28.3 months or the second bulb lasts for less than 28.2 months.
A purchasing manager at a university is investigating which brand of LCD projector to purchase for equipping classrooms. Of major concern to her is the lifetime of the light bulbs used in the projectors. One company has published the following frequency distribution from a test of the lifetimes (in hours) of 45 bulbs used in its LCD projectors.Bulb lifetime(in hours) Frequency700 to 749 4750 to 799 8800 to 849 14850 to 899 10900 to 949 5950 to 999 4Based on the frequency distribution, using the midpoint of each data class, estimate the mean lifetime for the light bulbs in the company's test. For your intermediate computations, use four or more decimal places, and round your answer to one decimal place.hours
What is the electrical efficiency of a 100 Wlight bulb that produces 5.7 kJ (57 000 J) ofheat when it is on for 10 min? (Hint: remem-ber that the bulb is supposed to producelight.) Show your work
An electric bulb is marked with 220 V, 50 W. Find the resistance of the filamentof thebulb. If the bulb is used 5 hours daily, find the energy in kilowatt-hour consumed bythe bulb in one month (30 days
The following table gives the life time in days for 100 tubelights. Find the mean life time of tubes by assumed mean method.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.