Every chemical element goes through natural exponential decay, which means that over time its atoms fall apart. The speed of each element's decay is described by its half-life, which is the amount of time it takes for the number of radioactive atoms of this element to be reduced by half. The half-life of the isotope dubnium $-263$ is $29$ seconds. A sample of dubnium $-263$ was first measured to have $1024$ atoms. After $t$ seconds, there were only $32$ atoms of this isotope remaining- Write an equation in terms of $t$ that models the situation.

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Q. Every chemical element goes through natural exponential decay, which means that over time its atoms fall apart. The speed of each element's decay is described by its half-life, which is the amount of time it takes for the number of radioactive atoms of this element to be reduced by half. The half-life of the isotope dubnium

-263

29

seconds. A sample of dubnium

-263

was first measured to have

1024

atoms. After

t

seconds, there were only

32

atoms of this isotope remaining- Write an equation in terms of

t

that models the situation.

Identify Initial Amount: Identify the initial amount of dubnium $-263$ and the remaining amount after $t$ seconds. Use the half-life to set up the decay formula. $\newline$ Initial amount ( $N_0$ ) = $1024$ atoms, Remaining amount ( $N$ ) = $32$ atoms, Half-life ( $t_{1/2}$ ) = $29$ seconds.
Use Decay Formula: Use the exponential decay formula $N = N_0 \times (1/2)^{t/t_{1/2}}$ . Substitute the known values to find $t$ . $32 = 1024 \times (1/2)^{t/29}$
Simplify Equation: Simplify the equation by dividing both sides by $1024$ . $\newline$ $\frac{32}{1024} = \left(\frac{1}{2}\right)^{\frac{t}{29}}$
Calculate $32/1024$ : Calculate $32/1024$ . $\newline$ $32/1024 = 1/32$
Recognize Base: Recognize that $\frac{1}{32}$ is $2^{-5}$ . Rewrite the equation using this base. $\newline$ $2^{-5} = \left(\frac{1}{2}\right)^{\frac{t}{29}}$
Convert Equation: Convert the equation to have the same base for easier comparison. $\newline$ $2^{-5} = 2^{-\frac{t}{29}}$
Equate Exponents: Equate the exponents since the bases are the same. $\newline$ $-5 = -\frac{t}{29}$
Solve for $t$ : Solve for $t$ by multiplying both sides by $-29$ . $t = 145$