Question
The rate at which an office building consumes energy (in kilowatt-hours(KWh) per week) is approximated by \(f(t) = 31.164e^{0.7261t}\), where \(t\) is in weeks since February 1, 2020. Energy consumption costs 12 cents per kilowatt-hour.
Use three digit accuracy in all computations.
Select each of the true statements from the following:
Answerlist
- The rate at which energy is consumed 2 weeks after February 1, 2020 is \(f(1)\).
- The rate at which energy is consumed 4 weeks after February 1, 2020 is \(f(31.164)\).
- The total amount of energy used by the office building during the 3 weeks after February 1, 2020 is \(\int_{0}^{3} f(t)dt\).
- An antiderivative of \(f(t) = 31.164e^{0.7261t}\) is \(F(t) = 42.92e^{0.7261t}\)
- \(\int_{0}^{3} f(t)dt = 31.164e^{0.7261(3)}-31.164e^{0.7261(0)}\)
- The total amount of energy used by the office building during the 4 weeks after February 1, 2020 is 740.549 KWh.
- The energy cost for the office building during the 2 weeks after February 1, 2020 is $22.
- All of the above statements are false.
Solution
Answerlist
- False. The rate at which energy is consumed 2 weeks after February 1, 2020 is \(f(2) =\) 133.148 KWh per week.
- False. The rate at which energy is consumed 4 weeks after February 1, 2020 is \(f(4) =\) 568.877 KWh per week.
- True. The total amount of energy used by the office building during the 3 weeks after February 1, 2020 is \[\int_{0}^{3} 31.164e^{0.7261t}dt = \frac{31.164}{0.7261}e^{0.7261(3)}-\frac{31.164}{0.7261}e^{0.7261(0)}=336.116 \].
- True. An antiderivative of \(f(t) = 31.164e^{0.7261t}\) is \(F(t) = \frac{31.164}{0.7261}e^{0.7261t}=42.92e^{0.7261t}\)
- False. \[\int_{0}^{3} f(t)dt=\int_{0}^{3} 31.164e^{0.7261t}dt = \frac{31.164}{0.7261}e^{0.7261(3)}-\frac{31.164}{0.7261}e^{0.7261(0)}\].
- True. The total amount of energy used by the office building during the 4 weeks after February 1, 2020 is \[\int_{0}^{4} 31.164e^{0.7261t}dt = \frac{31.164}{0.7261}e^{0.7261(4)}-\frac{31.164}{0.7261}e^{0.7261(0)}=740.549 \text{ KWh}\].
- False. The total amount of energy used by the office building during the 2 weeks after February 1, 2020 is \[\int_{0}^{2} 31.164e^{0.7261t}dt = \frac{31.164}{0.7261}e^{0.7261(2)}-\frac{31.164}{0.7261}e^{0.7261(0)}=140.455 \text{ KWh}\] To find the energy cost during this period of time we simply multiply this total amount of energy by 0.12 dollars per KWh to get $16.85
- False.