Asked by Michael Lasorsa on May 10, 2024
Verified
After t years, the remaining mass y (in grams) of 24 grams of a radioactive elements whose half-life is 30 years is given by y=24(12) t/30,t≥0y = 24 \left( \frac { 1 } { 2 } \right) ^ { t / 30 } , t \geq 0y=24(21) t/30,t≥0 . How much of the initial mass remains after 50 years? Round your answers to two decimal places.
A) 7.56 grams
B) 14.40 grams
C) 3.17 grams
D) 16.44 grams
E) nothing remains
Radioactive Elements
Elements that emit radiation as a result of the spontaneous disintegration of their atomic nuclei.
Half-Life
The time required for a quantity to reduce to half its initial value, often used in the context of radioactive decay.
\(24\left(\frac{1}{2}\right)^{\frac{t}{30}}\)
An exponential function representing the decay or growth of a quantity, where 24 is the initial value, \(\frac{1}{2}\) is the base of the exponent signifying decay, and \(\frac{t}{30}\) is the exponent indicating time or another variable at which the decay occurs.
- Gain insight into the exponential growth and decay model.
- Use exponential functions to model real-world scenarios such as population growth and radioactive decay.
Verified Answer
GN
George NyakundiMay 15, 2024
Final Answer :
A
Explanation :
To find the remaining mass after 50 years, substitute t=50t = 50t=50 into the given formula: y=24(12)50/30y = 24 \left( \frac { 1 } { 2 } \right) ^ { 50 / 30 }y=24(21)50/30 . This simplifies to y=24(12)5/3y = 24 \left( \frac { 1 } { 2 } \right) ^ { 5 / 3 }y=24(21)5/3 , which calculates to approximately 7.56 grams.
Learning Objectives
- Gain insight into the exponential growth and decay model.
- Use exponential functions to model real-world scenarios such as population growth and radioactive decay.