Half-life of Radioactive Substance: Solve Algebraically w/Logarithms

[Z-Score]

If 3150g of a radioactive substance decays to 450g in 73 weeks, determine the half life of the substance to the nearest week. Solve algebraically using logarithms.

450=3150(x)^73
450/3150=x^73
73sqrt(450/3150)=x

Now, the only problem is the answer says it is 26 weeks. Any help is much appreciated.

 

[Tide]

You're starting with the wrong equation. Try something like

$$450 = 3150 \left( \frac {1}{2} \right)^{73/T}$$

where T is the half-life.

 

[Architeuthis Dux]

To solve this problem algebraically, we can use the formula for exponential decay:

N(t) = N0 * (1/2)^(t/T)

Where N(t) is the amount of substance remaining after time t, N0 is the initial amount of substance, and T is the half-life of the substance.

In this case, we know that N(t) = 450g, N0 = 3150g, and t = 73 weeks. Plugging in these values, we get:

450 = 3150 * (1/2)^(73/T)

Dividing both sides by 3150, we get:

0.143 = (1/2)^(73/T)

To solve for T, we can take the logarithm of both sides with base 2:

log2(0.143) = log2((1/2)^(73/T))

Using the power rule of logarithms, we get:

log2(0.143) = (73/T) * log2(1/2)

Simplifying, we get:

log2(0.143) = -(73/T)

Rearranging, we get:

T = -73/log2(0.143)

Using a calculator, we find that log2(0.143) ≈ -2.86. Substituting this value into the equation, we get:

T ≈ -73/(-2.86) ≈ 25.52

Therefore, the half-life of this radioactive substance is approximately 25.52 weeks. Since we are asked to round to the nearest week, the half-life is 26 weeks.

It is possible that the answer given in the problem was rounded from the actual value of 25.52 weeks. However, it is also important to note that the half-life of a substance can vary slightly due to factors such as temperature and pressure. So, there may be a slight discrepancy between the calculated value and the given answer.