Suppose that an old hospital building leaks N2O (nitrous oxide) into the air. It leaks continuously...

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Suppose that an old hospital building leaks N2O (nitrous oxide) into the air. It leaks continuously (all the time), but not at a constant rate. Suppose A(t) gives the total volume of N2O (measured in cubic metres) that the hospital has leaked since midnight on 1 March, 2023, which is t = 0. Time, t, is measured in weeks and the domain of A(t) is D = [0, ). The table below shows some select values of A(t): t: 12345 6 7 8 9 10 11 12 A(t): 0.5 1.3 1.9 2.9 4.2 5.6 6.4 8.7 10.6 11.6 12.3 14.4 1. (1 mark) How much N2O did the hospital leak from midnight on 15th March 2023 until midnight on 5th April 2023? Show your work. 2. (a) (2 marks) Is A(t) an invertible function? Explain. (b) (1 mark) Is A-1(10.6) defined? If yes, find it's value (either exactly or approximately) and explain with a sentence what it represents. If not, explain why not. (c) (1 mark) Is A1 (10.8) defined? Explain. 3. Now consider the function N(t), which gives the volume of N2O (measured in cubic metres) leaked by the hospital so far during week t. (Week 1 would be from midnight on 1 March, 2023 until midnight on 8 March, 2023.) Time, t, is measured in weeks and the domain of N(t) is D = {1,2,3,4,...} (only positive integers). (a) (1 mark) Find N(7). (b) (2 marks) Is N(t) an invertible function? Explain. 4. (2 marks) Suppose that the same hospital also leaks desflurane (another greenhouse gas sometimes used in hospitals). After collecting some data, a team of scientists decides to model the desflurane leaked by: 4 ln(t+3)-2 D(t) = In(t+3) where D(t) gives the total volume of desflurane (measured in cubic metres) that the hospital has leaked since midnight on 1 March, 2023. Time, t, is measured in weeks and the domain of D(t) is D = [0, ). At what time, t, is the total volume of leaked desflurane 3 cubic metres? Give both an exact answer AND an answer rounded to 2 decimal places. Suppose that an old hospital building leaks N2O (nitrous oxide) into the air. It leaks continuously (all the time), but not at a constant rate. Suppose A(t) gives the total volume of N2O (measured in cubic metres) that the hospital has leaked since midnight on 1 March, 2023, which is t = 0. Time, t, is measured in weeks and the domain of A(t) is D = [0, ). The table below shows some select values of A(t): t: 12345 6 7 8 9 10 11 12 A(t): 0.5 1.3 1.9 2.9 4.2 5.6 6.4 8.7 10.6 11.6 12.3 14.4 1. (1 mark) How much N2O did the hospital leak from midnight on 15th March 2023 until midnight on 5th April 2023? Show your work. 2. (a) (2 marks) Is A(t) an invertible function? Explain. (b) (1 mark) Is A-1(10.6) defined? If yes, find it's value (either exactly or approximately) and explain with a sentence what it represents. If not, explain why not. (c) (1 mark) Is A1 (10.8) defined? Explain. 3. Now consider the function N(t), which gives the volume of N2O (measured in cubic metres) leaked by the hospital so far during week t. (Week 1 would be from midnight on 1 March, 2023 until midnight on 8 March, 2023.) Time, t, is measured in weeks and the domain of N(t) is D = {1,2,3,4,...} (only positive integers). (a) (1 mark) Find N(7). (b) (2 marks) Is N(t) an invertible function? Explain. 4. (2 marks) Suppose that the same hospital also leaks desflurane (another greenhouse gas sometimes used in hospitals). After collecting some data, a team of scientists decides to model the desflurane leaked by: 4 ln(t+3)-2 D(t) = In(t+3) where D(t) gives the total volume of desflurane (measured in cubic metres) that the hospital has leaked since midnight on 1 March, 2023. Time, t, is measured in weeks and the domain of D(t) is D = [0, ). At what time, t, is the total volume of leaked desflurane 3 cubic metres? Give both an exact answer AND an answer rounded to 2 decimal places.

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