It would be 2,112 yards = 1.2 miles
Answer:
4 1/4
Step-by-step explanation:
2 scoops would give you 3 cups. one more will give you 4 and a half so the baker would need 1/4 more
Answer:
(a) ![A = A_0 * e^{kt}](https://tex.z-dn.net/?f=A%20%3D%20A_0%20%2A%20e%5E%7Bkt%7D)
(b) There will be 1lb left after 14 hours
Step-by-step explanation:
Solving (a): The equation
Since the substance decomposes at a proportional rate, then it follows the following equation
![A(t) = A_0 * e^{kt}](https://tex.z-dn.net/?f=A%28t%29%20%3D%20A_0%20%2A%20e%5E%7Bkt%7D)
Where
Initial Amount
rate
time
Amount at time t
Solving (b):
We have:
![t = 4.6hr](https://tex.z-dn.net/?f=t%20%3D%204.6hr)
![A_0 = 8](https://tex.z-dn.net/?f=A_0%20%3D%208)
![A(4.6) = 4](https://tex.z-dn.net/?f=A%284.6%29%20%3D%204)
First, we calculate k using:
![A(t) = A_0 * e^{kt}](https://tex.z-dn.net/?f=A%28t%29%20%3D%20A_0%20%2A%20e%5E%7Bkt%7D)
This gives:
![A(4.6) = 8 * e^{k*4.6}](https://tex.z-dn.net/?f=A%284.6%29%20%3D%208%20%2A%20e%5E%7Bk%2A4.6%7D)
Substitute: ![A(4.6) = 4](https://tex.z-dn.net/?f=A%284.6%29%20%3D%204)
![4 = 8 * e^{k*4.6}](https://tex.z-dn.net/?f=4%20%3D%208%20%2A%20e%5E%7Bk%2A4.6%7D)
Divide both sides by 4
![0.5 = e^{k*4.6}](https://tex.z-dn.net/?f=0.5%20%3D%20e%5E%7Bk%2A4.6%7D)
Take natural logarithm of both sides
![\ln(0.5) = \ln(e^{k*4.6})](https://tex.z-dn.net/?f=%5Cln%280.5%29%20%3D%20%5Cln%28e%5E%7Bk%2A4.6%7D%29)
This gives:
![-0.6931 = k*4.6](https://tex.z-dn.net/?f=-0.6931%20%3D%20k%2A4.6)
Solve for k
![k = \frac{-0.6931}{4.6}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B-0.6931%7D%7B4.6%7D)
![k = -0.1507](https://tex.z-dn.net/?f=k%20%3D%20-0.1507)
So, we have:
![A(t) = A_0 * e^{kt}](https://tex.z-dn.net/?f=A%28t%29%20%3D%20A_0%20%2A%20e%5E%7Bkt%7D)
![A(t) = 8e^{-0.1507t}](https://tex.z-dn.net/?f=A%28t%29%20%3D%208e%5E%7B-0.1507t%7D)
To calculate the time when 1 lb will remain, we have:
![A(t) = 1](https://tex.z-dn.net/?f=A%28t%29%20%3D%201)
So, the equation becomes
![1= 8e^{-0.1507t}](https://tex.z-dn.net/?f=1%3D%208e%5E%7B-0.1507t%7D)
Divide both sides by 8
![0.125= e^{-0.1507t}](https://tex.z-dn.net/?f=0.125%3D%20e%5E%7B-0.1507t%7D)
Take natural logarithm of both sides
![\ln(0.125)= \ln(e^{-0.1507t})](https://tex.z-dn.net/?f=%5Cln%280.125%29%3D%20%5Cln%28e%5E%7B-0.1507t%7D%29)
![-2.0794= -0.1507t](https://tex.z-dn.net/?f=-2.0794%3D%20-0.1507t)
Solve for t
![t = \frac{-2.0794}{-0.1507}](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7B-2.0794%7D%7B-0.1507%7D)
![t = 13.7983](https://tex.z-dn.net/?f=t%20%3D%2013.7983)
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