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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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. Initial amount (N₀) = 1024 atoms, Remaining amount (N) = 32 atoms, Half-life (t₁/₂) = 29 seconds.Use Decay Formula:  Use the exponential decay formula N = N₀ * (1/2)^(t/t₁/₂). Substitute the known values to find t. 32 = 1024 * (1/2)^(t/29)Simplify Equation:  Simplify the equation by dividing both sides by 1024. 32/1024 = (1/2)^(t/29)Calculate 32/1024:  Calculate 32/1024. 32/1024 = 1/32Recognize Base:  Recognize that 1/32 is 2^(-5). Rewrite the equation using this base. 2^(-5) = (1/2)^(t/29)Convert Equation:  Convert the equation to have the same base for easier comparison. 2^(-5) = 2^(-t/29)Equate Exponents:  Equate the exponents since the bases are the same. -5 = -t/29Solve for t:  Solve for t by multiplying both sides by -29. t = 145

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