Question

1. Let

y
represent the mass of a quantity of a radioactive element whose half-life is 25 years. After
t

years, the mass (in grams) is

25
2
1
10
t
y 





 .
a. What is the initial mass of the quantity?
b. How much of the initial mass is present after 80 years?
c. After how many years will only 3 grams be present?

Answer 1

a. To find the initial mass of the quantity, we need to substitute t = 0 into the given equation.

1. Substitute t = 0 into the equation:

y = (25/2)^(0)

y = 25/2

y = 12.5

Therefore, the initial mass of the quantity is 12.5 grams.

b. To find how much of the initial mass is present after 80 years, we need to substitute t = 80 into the given equation.

1. Substitute t = 80 into the equation:

y = (25/2)^(80)

y ≈ 0.00057

Therefore, after 80 years, approximately 0.00057 grams of the initial mass is present.

c. To find after how many years only 3 grams will be present, we need to solve the given equation for t.

1. Set y = 3 in the equation:

3 = (25/2)^t

2. Take the logarithm of both sides (base doesn't matter):

log(3) = log((25/2)^t)

3. Apply the logarithm property to bring down the exponent:

log(3) = t * log(25/2)

4. Divide both sides by log(25/2):

t = log(3) / log(25/2)

5. Use a calculator to approximate the value of t:

t ≈ 13.6 years

Therefore, after approximately 13.6 years, only 3 grams will be present.