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 beryllium-11 is 14 seconds. A sample of beryllium- 11 was first measured to have 800 atoms. After t seconds, there were only 50 atoms of this isotope remaining. Write an equation in terms of t that models the situation.

Question

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 beryllium-11 is 14 seconds. A sample of beryllium- 11 was first measured to have 800 atoms. After  t seconds, there were only 50 atoms of this isotope remaining. Write an equation in terms of  t that models the situation.

Bytelearn · Accepted Answer

Rephrase the Question: First, let's rephrase the  ＂What is the exponential decay equation that models the reduction of beryllium-11 atoms from 800 to 50 over time t, given its half-life of 14 seconds?＂Identify Quantities: Identify the initial quantity (a) and the remaining quantity (y) after time t. Initial quantity (a): 800 atoms Remaining quantity (y): 50 atomsDetermine Half-life: Determine the half-life (h) of the substance. Half-life (h) of beryllium-11: 14 secondsCalculate Decay Factor: Since the half-life is the time it takes for a substance to reduce to half its initial amount, the decay factor (b) can be calculated using the formula b = 1/2^(1/h), where h is the half-life. Calculate the decay factor (b) for beryllium-11. b = 1/2^(1/14)Write Decay Equation: Perform the calculation for the decay factor (b). b = 1/2^(1/14) ≈ 0.9496 (rounded to four decimal places for simplicity)Solve for Remaining Atoms: Now, we can write the exponential decay equation in the form y = a(b)^t, where y is the remaining quantity after time t, a is the initial quantity, and b is the decay factor. Substitute the values for a and b into the equation. y = 800(0.9496)^tTake Natural Logarithm: To find the time t when there are only 50 atoms remaining, we need to solve the equation 50 = 800(0.9496)^t. Take the natural logarithm of both sides to solve for t. ln(50) = ln(800(0.9496)^t)Simplify Equation: Apply the properties of logarithms to simplify the equation. ln(50) = ln(800) + t*ln(0.9496)Isolate Variable: Isolate t on one side of the equation. t = (ln(50) - ln(800)) / ln(0.9496)Calculate Time: Perform the calculation to find the value of t. t ≈ (ln(50) - ln(800)) / ln(0.9496) t ≈ (3.912 - 6.685) / -0.0513 t ≈ -2.773 / -0.0513 t ≈ 54.06 seconds

Full solution

More problems from Write exponential functions: word problems

Related topics