Math  /  Algebra

QuestionElement X is a radioactive isotope such that every 25 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 50 grams, how much of the element would remain after 23 years, to the nearest whole number?

Studdy Solution

STEP 1

1. Element X is a radioactive isotope with a half-life of 25 years.
2. The initial mass of the sample is 50 grams.
3. We need to find the remaining mass after 23 years.
4. The decay follows an exponential decay model.

STEP 2

1. Understand the exponential decay formula.
2. Identify the parameters for the decay formula.
3. Substitute the known values into the formula.
4. Calculate the remaining mass and round to the nearest whole number.

STEP 3

Understand the exponential decay formula. The formula for exponential decay is given by:
M(t)=M0(12)tT M(t) = M_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}
where: - M(t) M(t) is the mass remaining after time t t , - M0 M_0 is the initial mass, - T T is the half-life period, - t t is the elapsed time.

STEP 4

Identify the parameters for the decay formula. From the problem, we have:
- M0=50 M_0 = 50 grams, - T=25 T = 25 years, - t=23 t = 23 years.

STEP 5

Substitute the known values into the formula:
M(23)=50(12)2325 M(23) = 50 \left(\frac{1}{2}\right)^{\frac{23}{25}}

STEP 6

Calculate the remaining mass:
First, calculate the exponent:
23250.92 \frac{23}{25} \approx 0.92
Now, calculate the decay factor:
(12)0.920.535 \left(\frac{1}{2}\right)^{0.92} \approx 0.535
Multiply by the initial mass:
M(23)=50×0.53526.75 M(23) = 50 \times 0.535 \approx 26.75
Round to the nearest whole number:
The remaining mass is approximately 27 \boxed{27} grams.

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