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.