What should be a simple integral: f is exponential fcn, T_{1⁄2}= 7. f(3)=12

[LM§]

After over 10 years I have to repeat all the high school maths in 15 days. It's a bit of a struggle

The following is giving me big problems, although I think it should be pretty simple and to the point:

f is an exponential function and T_1⁄2 equals 7.
f(3) = 12.

a) f(-11) = ?
b) f(10) = ?
c) solve the equation f(x) = 3/2.

 

[Deleted member 4993]

After over 10 years I have to repeat all the high school maths in 15 days. It's a bit of a struggle

The following is giving me big problems, although I think it should be pretty simple and to the point:

f is an exponential function and T_1⁄2 equals 7.
f(3) = 12.

a) f(-11) = ?
b) f(10) = ?
c) solve the equation f(x) = 3/2.

As posted, your problem is not making sense to me!!

You write:
and T_1⁄2 equals 7

What is T? How is 1/2 related to that?

 

[Dr.Peterson]

After over 10 years I have to repeat all the high school maths in 15 days. It's a bit of a struggle

The following is giving me big problems, although I think it should be pretty simple and to the point:

f is an exponential function and T_1⁄2 equals 7.
f(3) = 12.

a) f(-11) = ?
b) f(10) = ?
c) solve the equation f(x) = 3/2.

I presume you are using T_1⁄2 to mean half-life, and that this is exponential decay, not growth? But there is no integration involved, unless you are starting with the differential equation.

There are a couple ways to handle this, depending on how you have learned to write an exponential function. The easiest in terms of half-life is A = A_o(2)^t/T, where I'm just using T for half-life. It's just a little harder if you use base e, or something like (1+r) as the base.

You could just plug in t=3 and A=12 to find A_o, then evaluate for each question.

If you need additional help, please show some work, at least to give us an idea of how you are thinking of the equation. Ideally, show enough work so we can see what "big problems" you are having, and focus our attention on what you need most.

 

[HallsofIvy]

I presume you are using T_1⁄2 to mean half-life, and that this is exponential decay, not growth? But there is no integration involved, unless you are starting with the differential equation.

There are a couple ways to handle this, depending on how you have learned to write an exponential function. The easiest in terms of half-life is A = A_o(2)^t/T, where I'm just using T for half-life. It's just a little harder if you use base e, or something like (1+r) as the base.

With "decay" and "half life" that would be
A = A_o(1/2)^t/T or
A = A_o(2)^-t/T

You could just plug in t=3 and A=12 to find A_o, then evaluate for each question.

If you need additional help, please show some work, at least to give us an idea of how you are thinking of the equation. Ideally, show enough work so we can see what "big problems" you are having, and focus our attention on what you need most.

 

[LM§]

I presume you are using T_1⁄2 to mean half-life, and that this is exponential decay, not growth? But there is no integration involved, unless you are starting with the differential equation.

There are a couple ways to handle this, depending on how you have learned to write an exponential function. The easiest in terms of half-life is A = A_o(2)^t/T, where I'm just using T for half-life. It's just a little harder if you use base e, or something like (1+r) as the base.

You could just plug in t=3 and A=12 to find A_o, then evaluate for each question.

If you need additional help, please show some work, at least to give us an idea of how you are thinking of the equation. Ideally, show enough work so we can see what "big problems" you are having, and focus our attention on what you need most.

Yes, with T_1⁄2 I mean half-life. And the function would be f(x) = a^x.

From the information I was given, I can see that T_1⁄2 =ln(1/2)/ln(a). From there I get that ln(a) = ln(1/2)/T_1⁄2, and get ln(a) = -0,1. And that's how far I get. I am having problems finding the a.

 

[Deleted member 4993]

Yes, with T_1⁄2 I mean half-life. And the function would be f(x) = a^x.

From the information I was given, I can see that T_1⁄2 =ln(1/2)/ln(a). From there I get that ln(a) = ln(1/2)/T_1⁄2, and get ln(a) = -0,1. And that's how far I get. I am having problems finding the a.

Use the properties of logarithm:

If ln(x) = y then x = e^y

 

[LM§]

Use the properties of logarithm:

If ln(x) = y then x = e^y

I did all that and got results:

a = 0,9

f(-11) = a^(-11) = 3,18
f(10) = a^10 = 0,34.

When I check the results I got from the university it looks like it's all wrong. Their results are

f(-11) = 48
f(10) = 6.

What am I doing wrong?

 

[stapel]

I did all that and got results:

a = 0,9

Where did "a" come from?

f(-11) = a^(-11) = 3,18
f(10) = a^10 = 0,34.

Are you assuming that the equation is of the form "f(t) = a^{kt}", with no multiplier, such as "f(t) = P_0 * a^{kt}"?

Please show your work and reasoning, including the form of equation (especially if you've been given a particular form which you are required to use). Thank you!

 

[Dr.Peterson]

I did all that and got results:

a = 0,9

f(-11) = a^(-11) = 3,18
f(10) = a^10 = 0,34.

When I check the results I got from the university it looks like it's all wrong. Their results are

f(-11) = 48
f(10) = 6.

What am I doing wrong?

As has been said, you should restate the entire problem, including the form of the equation, as we have gradually drawn it out of you, so we can more easily follow it. I think it is this:

f is an exponential function of the form f(x) = A a^x, and the half-life T_1⁄2 equals 7.
f(3) = 12.

a) f(-11) = ?
b) f(10) = ?
c) solve the equation f(x) = 3/2.
​

As I understand it, you've correctly found that ln(a) = -0.1, so that a = 0.90 (to the nearest hundredth). So your equation is f(x) = A 0.90^x. You know that f(3) = 12. Can you now find A?