Let
![T(t)](https://tex.z-dn.net/?f=T%28t%29)
be the temperature of the body
![t](https://tex.z-dn.net/?f=t)
hours after 1:30 PM. Then
![T(0) = 32.5](https://tex.z-dn.net/?f=T%280%29%20%3D%2032.5)
and
![T(1) = 30.3](https://tex.z-dn.net/?f=T%281%29%20%3D%2030.3)
.
Using Newton's Law of Cooling,
![\dfrac{dT}{dt} = k(T - T_s)](https://tex.z-dn.net/?f=%5Cdfrac%7BdT%7D%7Bdt%7D%20%3D%20k%28T%20-%20T_s%29)
, we have
![\dfrac{dT}{dt} = k(T-20)](https://tex.z-dn.net/?f=%5Cdfrac%7BdT%7D%7Bdt%7D%20%3D%20k%28T-20%29)
. Now let
![y = T - 20](https://tex.z-dn.net/?f=y%20%3D%20T%20-%2020)
, so
![y(0) = T(0) - 20 = 32.5 - 20 = 12.5](https://tex.z-dn.net/?f=y%280%29%20%3D%20T%280%29%20-%2020%20%3D%2032.5%20-%2020%20%3D%2012.5)
, so
![y](https://tex.z-dn.net/?f=y)
is a solution to the initial value problem
![dy/dt = ky](https://tex.z-dn.net/?f=dy%2Fdt%20%3D%20ky)
with
![y(0) = 12.5](https://tex.z-dn.net/?f=y%280%29%20%3D%2012.5)
.
By separating and integrating, we have
![y(t) = y(0)e^{kt} = 12.5e^{kt}](https://tex.z-dn.net/?f=y%28t%29%20%3D%20y%280%29e%5E%7Bkt%7D%20%3D%2012.5e%5E%7Bkt%7D)
.
![y(1) = 30.3 - 20\ \Rightarrow\ 12.5e^{k(1)} = e^{k} = \frac{10.3}{12.5}\ \Rightarrow \\ k = \ln \frac{10.3}{12.5} ](https://tex.z-dn.net/?f=y%281%29%20%3D%2030.3%20-%2020%5C%20%5CRightarrow%5C%20%2012.5e%5E%7Bk%281%29%7D%20%3D%20e%5E%7Bk%7D%20%3D%20%5Cfrac%7B10.3%7D%7B12.5%7D%5C%20%5CRightarrow%20%5C%5C%0Ak%20%3D%20%5Cln%20%5Cfrac%7B10.3%7D%7B12.5%7D%0A)
![y(t) = 37 - 20\ \Rightarrow\ 12.5e^{kt} = 17\ \Rightarrow\ e^{kt} = \frac{10.3}{12.5} \ \Rightarrow \\ \\ kt = \ln \frac{10.3}{12.5}\ \Rightarrow\ t = \left( \ln \frac{17}{12.5} \right)/ \frac{10.3}{12.5} \approx -1.588 \mathrm{\ h}](https://tex.z-dn.net/?f=y%28t%29%20%3D%2037%20-%2020%5C%20%5CRightarrow%5C%2012.5e%5E%7Bkt%7D%20%3D%2017%5C%20%5CRightarrow%5C%20e%5E%7Bkt%7D%20%3D%20%5Cfrac%7B10.3%7D%7B12.5%7D%20%5C%20%5CRightarrow%20%5C%5C%20%5C%5C%0Akt%20%3D%20%5Cln%20%5Cfrac%7B10.3%7D%7B12.5%7D%5C%20%5CRightarrow%5C%20t%20%3D%20%5Cleft%28%20%5Cln%20%5Cfrac%7B17%7D%7B12.5%7D%20%20%5Cright%29%2F%20%5Cfrac%7B10.3%7D%7B12.5%7D%20%20%5Capprox%20-1.588%20%5Cmathrm%7B%5C%20h%7D)
≈ 95 minutes. Thus the murder took place about 95 minutes before 1:30 PM, or 11:55 AM.
Answer:
a)220399212
b)448
Step-by-step explanation:
Answer:
Step-by-step explanation:
Si haces una fracción buscando su denominador te da resultados
Answer:
(8, -8)
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Equality Properties
- Multiplication Property of Equality
- Division Property of Equality
- Addition Property of Equality
- Subtraction Property of Equality
<u>Algebra I</u>
- Terms/Coefficients
- Coordinates (x, y)
- Solving systems of equations using substitution/elimination
Step-by-step explanation:
<u>Step 1: Define Systems</u>
y = x - 16
5y = 2x - 56
<u>Step 2: Solve for </u><em><u>x</u></em>
- Substitute in <em>y</em>: 5(x - 16) = 2x - 56
- Distribute 5: 5x - 80 = 2x - 56
- [Subtraction Property of Equality] Subtract 2x on both sides: 3x - 80 = -56
- [Addition Property of Equality] Add 80 on both sides: 3x = 24
- [Division Property of Equality] Divide 3 on both sides: x = 8
<u>Step 3: Solve for </u><em><u>y</u></em>
- Define original equation: y = x - 16
- Substitute in <em>x</em>: y = 8 - 16
- Subtract: y = -8
Answer:
E
Step-by-step explanation:
6^2-5+4/5*1
36-5+4/5*1
36-5+0.8*1
36-5+0.8
31+0.8
31.8