Questions: A sample of a radioactive substance has an initial mass of 62.3 mg. This substance follows a continuous exponential decay model and has a half-life of 22 hours. (a) Let t be the time (in hours) since the start of the experiment, and let y be the amount of the substance at time t. Write a formula relating y to t. Use exact expressions to fill in the missing parts of the formula. Do not use approximations. y= 62.3 * (1/2)^(t/22) (b) How much will be present in 13 hours? Do not round any intermediate computations, and round your answer to the nearest tenth. 41.4 mg

A sample of a radioactive substance has an initial mass of 62.3 mg. This substance follows a continuous exponential decay model and has a half-life of 22 hours.
(a) Let t be the time (in hours) since the start of the experiment, and let y be the amount of the substance at time t.

Write a formula relating y to t.
Use exact expressions to fill in the missing parts of the formula.
Do not use approximations.

y= 62.3 * (1/2)^(t/22)

(b) How much will be present in 13 hours?

Do not round any intermediate computations, and round your answer to the nearest tenth.
41.4 mg

Transcript text: A sample of a radioactive substance has an initial mass of 62.3 mg. This substance follows a continuous exponential decay model and has a half-life of 22 hours. (a) Let $t$ be the time (in hours) since the start of the experiment, and let $y$ be the amount of the substance at time $t$. Write a formula relating $y$ to $t$. Use exact expressions to fill in the missing parts of the formula. Do not use approximations. \[ y=\square \prod^{(\square)} t \] (b) How much will be present in 13 hours? Do not round any intermediate computations, and round your answer to the nearest tenth. 41.4 mg

Solution

Solution Steps

Step 1: Calculate the Decay Constant ($\lambda$)

The decay constant ($\lambda$) is calculated using the formula: \[ \lambda = \frac{\ln(2)}{T_{1/2}} \] Substituting the given half-life ($T_{1/2} = 22$) into the formula, we get: \[ \lambda = \frac{\ln(2)}{22} = 0.0315 \]

Step 2: Calculate the Amount of Substance Remaining ($y$)

Using the continuous exponential decay formula: \[ y = y_0 e^{-\lambda t} \] Substituting $y_0 = 62.3$, $\lambda = 0.0315$, and $t = 13$, we get: \[ y = 62.3 \times e^{-0.0315 \times 13} = 41.4 \]