Questions: Español A specific radioactive substance follows a continuous exponential decay model. It has a half-life of 14 minutes. At the start of the experiment, 84.6 g is present. (a) Let t be the time (in minutes) 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 = □ e^(□ t) (b) How much will be present in 23 minutes? Do not round any intermediate computations, and round your answer to the nearest tenth. □

Español
A specific radioactive substance follows a continuous exponential decay model. It has a half-life of 14 minutes. At the start of the experiment, 84.6 g is present.
(a) Let t be the time (in minutes) 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 = □ e^(□ t)

(b) How much will be present in 23 minutes?

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

Transcript text: Español A specific radioactive substance follows a continuous exponential decay model. It has a half-life of 14 minutes. At the start of the experiment, 84.6 g is present. (a) Let $t$ be the time (in minutes) 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. \[ y=\square e^{(\square) t} \] (b) How much will be present in 23 minutes? Do not round any intermediate computations, and round your answer to the nearest tenth. $\square$

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 ($T_{1/2} = 14$) into the formula, we get: \[ \lambda = \frac{\ln(2)}{14} = 0.0495 \]

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

Using the continuous exponential decay formula: \[ y = y_0 e^{-\lambda t} \] Substituting $y_0 = 84.6$, $\lambda = 0.0495$, and $t = 23$, we get: \[ y = 84.6 \times e^{-0.0495 \times 23} = 27.1 \]