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.
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.
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Solution
Solution Steps
To find the half-life of the radioactive goo, we can use the formula for exponential decay:
A(t)=A0⋅e−kt
where:
A(t) is the amount of substance remaining at time t
A0 is the initial amount of substance
k is the decay constant
t 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 t, we can use the formula G(t)=396⋅e−kt.
To find the half-life, we can set up the equation 49.5=396⋅e−240k and solve for k.
To find the amount of goo remaining after 26 minutes, we can use the formula G(26)=396⋅e−26k.
Step 1: Calculate the decay constant k
Given:
Initial amount A0=396
Final amount A(t)=49.5
Time elapsed for half-life t=240
Using the formula k=−tln(A(t)/A0), we find:
k=−240ln(49.5/396)≈0.0086643
Step 2: Calculate the amount of goo after 26 minutes
Given:
Time elapsed t=26
Using the formula G(t)=396⋅e−kt, we find:
G(26)=396⋅e−0.0086643⋅26≈316.13
Final Answer
The half-life of the goo is approximately 115.4 minutes. The amount of goo remaining after 26 minutes is approximately 316.1 grams.