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. $\newline$ 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. $\newline$ 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.

\newline

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.

\newline

Write an equation in terms of

t

that models the situation.

Exponential Decay Formula: The general formula for exponential decay is $N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}}$ , where $N(t)$ is the number of atoms at time $t$ , $N_0$ is the initial number of atoms, $\left(\frac{1}{2}\right)$ is the decay factor, and $T$ is the half-life of the substance.
Substitute Values: Given that the half-life $T$ of dubnium $-263$ is $29$ seconds, we can substitute this value into the formula. We also know that the initial number of atoms $N_0$ is $1024$ , and the remaining number of atoms $N(t)$ after time $t$ is $32$ .
Isolate Exponential Term: Substitute the given values into the decay formula: $32 = 1024 \times (1/2)^{(t/29)}$ .
Simplify Equation: To find the equation in terms of $t$ , we need to solve for $t$ . First, divide both sides of the equation by $1024$ to isolate the exponential term: $\frac{32}{1024} = \left(\frac{1}{2}\right)^{\frac{t}{29}}$ .
Recognize Exponential Form: Simplify the left side of the equation: $\frac{1}{32} = \left(\frac{1}{2}\right)^{\frac{t}{29}}$ .
Set Exponents Equal: Recognize that $\frac{1}{32}$ is $2^{-5}$ since $2^5 = 32$ . Therefore, we can rewrite the equation as $2^{-5} = \left(\frac{1}{2}\right)^{\frac{t}{29}}$ .
Solve for $t$ : Since $\frac{1}{2}$ is the same as $2^{-1}$ , we can rewrite the equation as $2^{-5} = 2^{-\frac{t}{29}}$ .
Calculate Final Value: By the property of exponents, if the bases are the same, then the exponents must be equal. Therefore, we can set the exponents equal to each other: $-5 = -\frac{t}{29}$ .
Calculate Final Value: By the property of exponents, if the bases are the same, then the exponents must be equal. Therefore, we can set the exponents equal to each other: $-5 = -\frac{t}{29}$ .Multiply both sides of the equation by $-29$ to solve for $t$ : $-5 \times -29 = t$ .
Calculate Final Value: By the property of exponents, if the bases are the same, then the exponents must be equal. Therefore, we can set the exponents equal to each other: $-5 = -\frac{t}{29}$ . Multiply both sides of the equation by $-29$ to solve for $t$ : $-5 \times -29 = t$ . Calculate the value of $t$ : $t = 145$ .