Questions: Homework 2.5: Exponential and Logarithmic Models Score: 15 / 100 Answered: 2 / 10 Question 3 At the beginning of an experiment, a scientist has 396 grams of radioactive goo. After 240 minutes, her sample has decayed to 49.5 grams. What is the half-life of the goo in minutes? Find a formula for G(t), the amount of goo remaining at time t. G(t)= How many grams of goo will remain after 26 minutes? You may enter the exact value or round to 2 decimal places.

Homework 2.5: Exponential and Logarithmic Models
Score: 15 / 100 Answered: 2 / 10

Question 3

At the beginning of an experiment, a scientist has 396 grams of radioactive goo. After 240 minutes, her sample has decayed to 49.5 grams.

What is the half-life of the goo in minutes? 

Find a formula for G(t), the amount of goo remaining at time t.

G(t)=

How many grams of goo will remain after 26 minutes? 
You may enter the exact value or round to 2 decimal places.
Transcript text: Homework 2.5: Exponential and Logarithmic Models Score: $15 / 100$ Answered: $2 / 10$ Question 3 At the beginning of an experiment, a scientist has 396 grams of radioactive goo. After 240 minutes, her sample has decayed to 49.5 grams. What is the half-life of the goo in minutes? $\square$ Find a formula for $G(t)$, the amount of goo remaining at time $t$. \[ G(t)= \] $\square$ How many grams of goo will remain after 26 minutes? $\square$ You may enter the exact value or round to 2 decimal places. Question Help: Video Submit Question
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Solution

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Solution Steps

To find the half-life of the radioactive goo, we can use the formula for exponential decay:

A(t)=A0ektA(t) = A_0 \cdot e^{-kt}

where:

  • A(t)A(t) is the amount of substance remaining at time tt
  • A0A_0 is the initial amount of substance
  • kk is the decay constant
  • tt is the time elapsed

We can use the given information to set up an equation to solve for the half-life.

To find the amount of goo remaining at time tt, we can use the formula G(t)=396ektG(t) = 396 \cdot e^{-kt}.

To find the half-life, we can set up the equation 49.5=396e240k49.5 = 396 \cdot e^{-240k} and solve for kk.

To find the amount of goo remaining after 26 minutes, we can use the formula G(26)=396e26kG(26) = 396 \cdot e^{-26k}.

Step 1: Calculate the decay constant k k

Given:

  • Initial amount A0=396 A_0 = 396
  • Final amount A(t)=49.5 A(t) = 49.5
  • Time elapsed for half-life t=240 t = 240

Using the formula k=ln(A(t)/A0)t k = -\frac{\ln(A(t) / A_0)}{t} , we find: k=ln(49.5/396)2400.0086643 k = -\frac{\ln(49.5 / 396)}{240} \approx 0.0086643

Step 2: Calculate the amount of goo after 26 minutes

Given:

  • Time elapsed t=26 t = 26

Using the formula G(t)=396ekt G(t) = 396 \cdot e^{-kt} , we find: G(26)=396e0.008664326316.13 G(26) = 396 \cdot e^{-0.0086643 \cdot 26} \approx 316.13

Final Answer

The half-life of the goo is approximately 115.4 \boxed{115.4} minutes. The amount of goo remaining after 26 minutes is approximately 316.1 \boxed{316.1} grams.

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