evitagimm9h

2022-11-06

${e}^{pt}$or $(1+p{)}^{t}$ What is the difference in modeling exponential growth and decay?

I would really like to better recognize, whilst the feature ${e}^{pt}$ is the "higher" desire and while (if at all) $(1+p{)}^{t}$ have to be used.

to provide a conventional example: Say we need to version radioactive decay of some detail A. allow t be in units of one half-life of A. Then $p=-\frac{1}{2}$

Now I'd say the standard approach to modeling this is via the function

$${A}_{1}(t)={A}_{0}\cdot {(1-\frac{1}{2})}^{t}.$$

On the other hand, if we approach this problem as an ODE, we can say that at any point t we want A(t) to decrease at a rate of half of its momentary amount:

$$\frac{d}{dt}{A}_{2}(t)=-\frac{1}{2}{A}_{2}(t)\text{},$$

which leads to the function

$${A}_{2}(t)={A}_{0}\cdot {e}^{-\frac{1}{2}t}.$$

But which approach would be "better" here? I think ${A}_{1}$ is much more commonly (if not exceptionally) used when it comes to modeling atomic decay. On the other hand, I know that

$$e=\underset{n\to \mathrm{\infty}}{lim}{(1+\frac{1}{n})}^{n}$$

which essentially means that the rate of change of ${A}_{2}$ is continously updated, while ${A}_{1}$ is updated discretely, right? That's the best way I can phrase it at the moment.

So to conclude: Does this mean that ${A}_{2}$ is always the "better", more accurate choice or are there situations where ${A}_{1}$ is actually "correct"?

I would really like to better recognize, whilst the feature ${e}^{pt}$ is the "higher" desire and while (if at all) $(1+p{)}^{t}$ have to be used.

to provide a conventional example: Say we need to version radioactive decay of some detail A. allow t be in units of one half-life of A. Then $p=-\frac{1}{2}$

Now I'd say the standard approach to modeling this is via the function

$${A}_{1}(t)={A}_{0}\cdot {(1-\frac{1}{2})}^{t}.$$

On the other hand, if we approach this problem as an ODE, we can say that at any point t we want A(t) to decrease at a rate of half of its momentary amount:

$$\frac{d}{dt}{A}_{2}(t)=-\frac{1}{2}{A}_{2}(t)\text{},$$

which leads to the function

$${A}_{2}(t)={A}_{0}\cdot {e}^{-\frac{1}{2}t}.$$

But which approach would be "better" here? I think ${A}_{1}$ is much more commonly (if not exceptionally) used when it comes to modeling atomic decay. On the other hand, I know that

$$e=\underset{n\to \mathrm{\infty}}{lim}{(1+\frac{1}{n})}^{n}$$

which essentially means that the rate of change of ${A}_{2}$ is continously updated, while ${A}_{1}$ is updated discretely, right? That's the best way I can phrase it at the moment.

So to conclude: Does this mean that ${A}_{2}$ is always the "better", more accurate choice or are there situations where ${A}_{1}$ is actually "correct"?

dilettato5t1

Beginner2022-11-07Added 25 answers

"It depends how large the decay period is. Let´s say the length of the decay period is $\frac{1}{n}$. Then in n periods the amount of units of the atom after t periods is

${A}_{t}={A}_{0}\cdot {(1-\frac{\frac{1}{2}}{n})}^{n\cdot t}$, where ${A}_{0}$ is the initial amount. Important: $\frac{1}{2}$ is the decay rate in a period of the legnth $\frac{1}{n}$.

The period can get smaller and smaller, which means that $n\to \mathrm{\infty}$.

$${A}_{t}={A}_{0}\cdot \underset{n\to \mathrm{\infty}}{lim}{(1-\frac{\frac{1}{2}}{n})}^{n\cdot t}={A}_{0}\cdot {e}^{-\frac{1}{2}\cdot t}$$

Conclusion: The smaller the decay/growth period gets the more it is reasonable to apply the continuous growth/decay model."

${A}_{t}={A}_{0}\cdot {(1-\frac{\frac{1}{2}}{n})}^{n\cdot t}$, where ${A}_{0}$ is the initial amount. Important: $\frac{1}{2}$ is the decay rate in a period of the legnth $\frac{1}{n}$.

The period can get smaller and smaller, which means that $n\to \mathrm{\infty}$.

$${A}_{t}={A}_{0}\cdot \underset{n\to \mathrm{\infty}}{lim}{(1-\frac{\frac{1}{2}}{n})}^{n\cdot t}={A}_{0}\cdot {e}^{-\frac{1}{2}\cdot t}$$

Conclusion: The smaller the decay/growth period gets the more it is reasonable to apply the continuous growth/decay model."

Find the volume V of the described solid S

A cap of a sphere with radius r and height h.

V=??

Whether each of these functions is a bijection from R to R.

a) $f(x)=-3x+4$

b) $f\left(x\right)=-3{x}^{2}+7$

c) $f(x)=\frac{x+1}{x+2}$

?

$d)f\left(x\right)={x}^{5}+1$In how many different orders can five runners finish a race if no ties are allowed???

State which of the following are linear functions?

a.$f(x)=3$

b.$g(x)=5-2x$

c.$h\left(x\right)=\frac{2}{x}+3$

d.$t(x)=5(x-2)$ Three ounces of cinnamon costs $2.40. If there are 16 ounces in 1 pound, how much does cinnamon cost per pound?

A square is also a

A)Rhombus;

B)Parallelogram;

C)Kite;

D)none of theseWhat is the order of the numbers from least to greatest.

$A=1.5\times {10}^{3}$,

$B=1.4\times {10}^{-1}$,

$C=2\times {10}^{3}$,

$D=1.4\times {10}^{-2}$Write the numerical value of $1.75\times {10}^{-3}$

Solve for y. 2y - 3 = 9

A)5;

B)4;

C)6;

D)3How to graph $y=\frac{1}{2}x-1$?

How to graph $y=2x+1$ using a table?

simplify $\sqrt{257}$

How to find the vertex of the parabola by completing the square ${x}^{2}-6x+8=y$?

There are 60 minutes in an hour. How many minutes are there in a day (24 hours)?

Write 18 thousand in scientific notation.