Answer:
The correct answer is:
Between 600 and 700 years (B)
Step-by-step explanation:
At a constant decay rate, the half-life of a radioactive substance is the time taken for the substance to decay to half of its original mass. The formula for radioactive exponential decay is given by:
![A(t) = A_0 e^{(kt)}\\where:\\A(t) = Amount\ left\ at\ time\ (t) = 75\ grams\\A_0 = initial\ amount = 1000\ grams\\k = decay\ constant\\t = time\ of\ decay = 2500\ years](https://tex.z-dn.net/?f=A%28t%29%20%3D%20A_0%20e%5E%7B%28kt%29%7D%5C%5Cwhere%3A%5C%5CA%28t%29%20%3D%20Amount%5C%20left%5C%20at%5C%20time%5C%20%28t%29%20%3D%2075%5C%20grams%5C%5CA_0%20%3D%20initial%5C%20amount%20%3D%201000%5C%20grams%5C%5Ck%20%3D%20decay%5C%20constant%5C%5Ct%20%3D%20time%5C%20of%5C%20decay%20%3D%202500%5C%20years)
First, let us calculate the decay constant (k)
![75 = 1000 e^{(k2500)}\\dividing\ both\ sides\ by\ 1000\\0.075 = e^{(2500k)}\\taking\ natural\ logarithm\ of\ both\ sides\\In 0.075 = In (e^{2500k})\\In 0.075 = 2500k\\k = \frac{In0.075}{2500}\\ k = \frac{-2.5903}{2500} \\k = - 0.001036](https://tex.z-dn.net/?f=75%20%3D%201000%20e%5E%7B%28k2500%29%7D%5C%5Cdividing%5C%20both%5C%20sides%5C%20by%5C%201000%5C%5C0.075%20%3D%20e%5E%7B%282500k%29%7D%5C%5Ctaking%5C%20natural%5C%20logarithm%5C%20of%5C%20both%5C%20sides%5C%5CIn%200.075%20%3D%20In%20%28e%5E%7B2500k%7D%29%5C%5CIn%200.075%20%3D%202500k%5C%5Ck%20%3D%20%5Cfrac%7BIn0.075%7D%7B2500%7D%5C%5C%20k%20%3D%20%5Cfrac%7B-2.5903%7D%7B2500%7D%20%5C%5Ck%20%3D%20-%200.001036)
Next, let us calculate the half-life as follows:
![\frac{1}{2} A_0 = A_0 e^{(-0.001036t)}\\Dividing\ both\ sides\ by\ A_0\\ \frac{1}{2} = e^{-0.001036t}\\taking\ natural\ logarithm\ of\ both\ sides\\In(0.5) = In (e^{-0.001036t})\\-0.6931 = -0.001036t\\t = \frac{-0.6931}{-0.001036} \\t = 669.02 years\\\therefore t\frac{1}{2} \approx 669\ years](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20A_0%20%3D%20A_0%20e%5E%7B%28-0.001036t%29%7D%5C%5CDividing%5C%20both%5C%20sides%5C%20by%5C%20A_0%5C%5C%20%5Cfrac%7B1%7D%7B2%7D%20%3D%20e%5E%7B-0.001036t%7D%5C%5Ctaking%5C%20natural%5C%20logarithm%5C%20of%5C%20both%5C%20sides%5C%5CIn%280.5%29%20%3D%20In%20%28e%5E%7B-0.001036t%7D%29%5C%5C-0.6931%20%3D%20-0.001036t%5C%5Ct%20%3D%20%5Cfrac%7B-0.6931%7D%7B-0.001036%7D%20%5C%5Ct%20%3D%20669.02%20years%5C%5C%5Ctherefore%20t%5Cfrac%7B1%7D%7B2%7D%20%20%5Capprox%20669%5C%20years)
Therefore the half-life is between 600 and 700 years
Answer:
Step-by-step explanation:
<u>Given</u>
- Each deposit = $1500
- Interest rate= 5%
- Time = 4 years
- Compound number = annual
<u>Simple interest account</u>
- B = 1500*(1 + 4*0.05) = 1500*1.2 = $1800
<u>Compound interest account</u>
- B = 1500*(1 + 0.05)^4 = $1823.26
<u>Total balance</u>
- $1800 + $1823.26 = $3623.26
Answer:
Step-by-step explanation:
25- guaranteed
$14 on each table waited on
He waited on 17.
17x14= $238 tips
238+25= $263
1. Marques expects to earn $263 on 17 tables
2. He expects to make 25+14t per day.
Answer:
Angle D= 140
Step-by-step explanation:
7x+7x+ 4x+4x+7x+7x= 720
36x=720
x= 20
Answer:
![\frac{1}{12}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B12%7D)
Step-by-step explanation:
P(tail) = ![\frac{1}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D)
P(2) = ![\frac{1}{6}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B6%7D)
P(tail and 2 ) =
×
= ![\frac{1}{12}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B12%7D)