#1. You want t where
e^(.0375t) = 2
Not E. What calculation did you use?
#2. Not A
since less than one half-life has passed, more than half of the original amount must remain. What calculation did you use?
A. 1.85 years
B. 1.93 years
C. 18.48 years
D. 18.83 years
E. 19.80 years*
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Cobalt-60 is used for radiotherapy. It has a half-life of 5.26 years. If 4 g of cobalt-60 is administered, how much remains in 3 years?
A. 1.2 g*
B. 2.7 g
C. 3.3 g
D. 2.1 g
E. 0.2 g
Could someone kindly help me double check these answers to these questions? Thank you all so much!!!!!! :) :D πβ»βΊπ γ :-)π
e^(.0375t) = 2
Not E. What calculation did you use?
#2. Not A
since less than one half-life has passed, more than half of the original amount must remain. What calculation did you use?
Question 1: To determine the amount of time it would take for an investment to double, we can use the formula for continuous compounding: A = P * e^(rt), where A is the final amount, P is the initial principal, r is the interest rate, and t is the time in years.
Let's set up the equation: 2 = e^(0.0375t), where 0.0375 is the interest rate of 3.75% in decimal form.
To solve for t, we can take the natural logarithm (ln) of both sides of the equation: ln(2) = ln(e^(0.0375t)).
Using the property of logarithms, we can bring down the exponent: ln(2) = 0.0375t * ln(e).
Since ln(e) is equal to 1, we can simplify the equation: 0.0375t = ln(2).
Now, we can solve for t: t = ln(2) / 0.0375 β 18.48 years.
Therefore, the correct answer is C: 18.48 years.
Question 2: The half-life of cobalt-60 is 5.26 years, which means that after each half-life, the amount remaining is reduced by half.
To find out how much remains in 3 years, we need to determine the number of half-lives that have passed. Dividing the time elapsed (3 years) by the half-life (5.26 years), we get 3 / 5.26 β 0.57.
Since half of the original amount remains after each half-life, we can calculate the remaining amount by multiplying the original amount (4 g) by 0.5 raised to the power of the number of half-lives: 4 * (0.5)^0.57 β 1.2 g.
Therefore, the correct answer is A: 1.2 g.
I hope that helps! Let me know if you have any more questions. :)
Question 1: Determine the amount of time it would take for an investment to double if interest is earned at a rate of 3.75% compounded continuously.
To find the amount of time it takes for an investment to double, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the final amount or future value
P = the initial amount or present value
e = the base of the natural logarithm, approximately 2.71828
r = the interest rate
t = time in years
In this case, since we want to find the time it takes for the investment to double, we can set A = 2P.
2P = P * e^(rt)
Next, we can cancel out the common factor of P:
2 = e^(rt)
To solve for t, we take the natural logarithm of both sides:
ln(2) = rt
Finally, we can solve for t by dividing both sides by r:
t = ln(2) / r
Substituting the given interest rate of 3.75% (or 0.0375 in decimal form) into the equation, we get:
t = ln(2) / 0.0375
We can use a calculator or graphing software to find the value of ln(2) / 0.0375. In this case, it is approximately 18.48 years.
Therefore, the correct answer is option C: 18.48 years.
Question 2: Cobalt-60 has a half-life of 5.26 years. If 4 g of cobalt-60 is administered, how much remains in 3 years?
To solve this question, we can use the formula for exponential decay:
N(t) = Nβ * (1/2)^(t / T)
Where:
N(t) = the amount remaining after time t
Nβ = the initial amount
t = time in years
T = the half-life of the substance
In this case, Nβ is given as 4 g, t is given as 3 years, and T is given as 5.26 years. Plugging these values into the formula, we get:
N(3) = 4 * (1/2)^(3 / 5.26)
Evaluating this expression, we find that N(3) is approximately 1.19 g.
Therefore, the amount of cobalt-60 remaining after 3 years is approximately 1.19 g.
The closest answer option to this value is option A: 1.2 g, which is correct.
I hope this helps! If you have any further questions, feel free to ask.