In a murder investigation, the temperature of the corpse
was $32.5^{\circ} \mathrm{C}$ at $1 : 30 \mathrm{PM}$ and $30.3^{\circ} \mathrm{C}$ an hour later. Normal
body temperature is $37.0^{\circ} \mathrm{C}$ and the temperature of the surroundings was $20.0^{\circ} \mathrm{C} .$ When did the murder take place?

Answer

$$
11 : 55 \mathrm{am}
$$

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Calculus 1 / AB

Essential Calculus Early Transcendentals

Chapter 3

INVERSE FUNCTIONS

Section 4

Exponential Growth and Decay

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Video Transcript

Hey, guys, Welcome back This problem we're getting on to use Newton's law of cooling, which states that the change in temperature over the change in time is you okay, multiplied by the quantity of temperature of the body minus the temperature of the surroundings. We know the tempter of surroundings are 20 degrees Celsius. We plug that in, we have D 80 over the little T deal. Okay. T minus 20. Make things simpler. We can let Why be equal to t minus 20 upon during that substitution, we end up the differential equation. Hello, Deepti. Over dtz. Okay. Why? Then, upon integrating that, we end up with this guy. Why's he go to our initial I value Times e raised the Katie. We want to let t equals zero the time 1 30 pm There are problems simpler and we know that at 1 30 corpse was 32.5 degrees, So ah, t equals zero. Our temperature was 32.5 degrees Celsius. But then we know that why equal to t minus 20 installed for why, if why? At T equals zero will be 32.5 minus 20 or why will be 12.5 degrees Celsius. We know that at 1 30 the temperature of the body was 30.3. I wouldn't say AT T go to one temperature of the body is 30.3 degrees Celsius. Again, we'll either substitution wise but the T minus 20. So why of 11 hour later? It should be 10 0.3 degrees Celsius. And I would have to equations in two unknowns first weekend user information here, greater first equation. And we have 12.5. Our initial y value is unknown and we'll have. He raised the K. He and our T in this case was zero, he said. Okay, so we know that we saw this out erased zero, just one. And we left with fact that Penis condition there's 12.5. That makes sense because it was at the T value that he said was considering to be zero at 1:30 p.m. Your initial condition is 12.5, which not use that effect that at T zero toe one temperatures 10.3. We can write that 10.3. It's legal to our initial condition. 12.5 he raised to the k Times t is just one. In this case, we can divide both sides by 12.5. We'll have 10.3 over 12.5 Siegel to e. Okay, we need the natural log of both sides and we get that K equal to the natural log 10.3 over 12 point five. Next, we want to realize that a normal body temperature is 37 degrees that state in the given. So the temperature of 37 degrees Celsius is a normal body temperature. We know that why is equal to T minus 20. We 37 minus 20 2017 degrees basically want to figure out when was why 17 degrees? And we know we have our equation. Why equal to why? Said Bo, 12.5 multiplied. But e race to the Katie. We know that this value is going to be negative. 0.19 358 with e raised the negative point 19 358 supplied by T. Now we just want to plug in 17 degrees for why? And figure out what tea is. We have 17 degrees equals 12.5 degrees times. He raced negative 0.19 358 1935 Okay, de divide both sides by 12.5 17. Divided by 12.5 B 1.36 We can take the natural log on both sides of natural log of one point 36 It's Eagle two Natural long that you will go away as well. Negative 0.19 358 Okay. And when you do that calculation your value of tea. Siegel too negative. One point five 88 Approximately. We have to realize, though, that that is in hours. So 1.588 hours it's just about one hour. One hour. It will have 0.588 minutes. Multiply that by 60 right around one hour and 35 minutes before we had our P equals zero. And we had previously set arbitrarily t equals zero to be, um, at 1 30 PM Do that over here. So if we are one hour and 35 minutes before T is equal to zero when I run 35 minutes before 1:30 p.m. That comes out that the murder took place at 11. 55 I am. And that is your final answer. Thanks for watching

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