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. Report a problem

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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. Report a problem

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Exponential Decay Formula: The general formula for exponential decay is N(t) = N_0 * (1/2)^(t/T), where N(t) is the number of atoms at time t, N_0 is the initial number of atoms, (1/2) 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 * (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: 32/1024 = (1/2)^(t/29).Recognize Exponential Form: Simplify the left side of the equation: 1/32 = (1/2)^(t/29).Set Exponents Equal: Recognize that 1/32 is 2^(-5) since 2^5 = 32. Therefore, we can rewrite the equation as 2^(-5) = (1/2)^(t/29).Solve for t: Since (1/2) is the same as 2^(-1), we can rewrite the equation as 2^(-5) = 2^(-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 = -t/29.Multiply both sides of the equation by -29 to solve for t: -5 * -29 = t.Calculate the value of t: t = 145.

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