Question

Solve for Y=Ae^kt

if Y=15.25 t=25
and Y=6.75 t=10

Find the value of A and K

Answer 1

Substitute values given for Y and t to get two equations:
\[15.25=Ae ^{25k}\]
\[6.75=Ae^{10k}\]
Hint: Now divide the first equation by the second equation:
\[\frac{15.25}{6.75}=\frac{Ae^{25k}}{Ae^{10k}}\]
Can you use exponent properties to solve for k? Once you know the value of k, you can substitute it back in either of the first two equations to find A.

Answer 2

what kind of problem are you doing

Answer 3

@ivycoveredwalls is it 2.22 = e^3K ?

Answer 4

I think that:
\[e ^{15k}=\frac{ 15.25 }{ 6.75 }\]
as you easily can check using the equation that @ivycoveredwalls has wrote

Answer 5

so going to logarithm of the above expression, we can write:
\[15*k=\ln(\frac{ 12.25 }{ 6.75 })\]
and finally:
\[k=\frac{ 1 }{ 15 }\ln(\frac{ 12.25 }{ 6.75 })\]

Answer 6

is the answer K equal to 1.7039129684 ?

Answer 7

Sorry, I don't think, please apply the above formula!

Answer 8

but why it is 12.25 above

Answer 9

because using the properties of the product of powers of the same base, we can write:
\[\frac{ 12.25 }{ 6.75 }=\frac{ e ^{25k} }{ e ^{10k} }=e ^{25k-10k}=e ^{15k}\]

Answer 10

I think it is 15.25 instead of 12.25 ?

Answer 11

Sorry, you are right!

Answer 12

then the K is 0.0594665359?

Answer 13

@Kyle` that's right!

Answer 14

Sorry I got k=0.0543

Answer 15

which one is correct :p?

Answer 16

I think k=0.0543

Answer 17

how about A !?

Answer 18

In order to find A, we can use, for example, the second equation which @ivycoveredwalls has wrote, namely:
\[6.75=A*e ^{10k}\]
Now, we can write, using the properties of powers with the same base:
\[e ^{10k}=(e ^{15k})^{2/3}\]
because:
\[15k*\frac{ 2 }{ 3 }=10k\]
Since:
\[e ^{15k}=\frac{ 15.25 }{ 6.75 }\]
then, we can write:
\[e ^{10k}=(\frac{ 15.25 }{ 6.75 })^{2/3}=1.72177\]
finally we get for A the subsequent value:
\[A=\frac{ 6.75 }{ 1.72 }\approx 3.92\]

Answer 19

what is the 2/3 ?

Answer 20

2/3=0.666666....

Answer 21

where is the 0.6666 come from !?

Answer 22

because to use the formula:
6.75=A*e^(10k), you have to calculate what is e^(10k).
Now, we have e^10k), nothe that being:
\[10k=\frac{ 2 }{ 3 }*15k\]
we can get e^(10k) simply performing the subsequent calculus:
\[e ^{10k}=(e ^{15k})^{(2/3)}\]

Answer 23

since, applying the rule of the power of power, we can write:
\[(e ^{15k})^{(2/3)}=e ^{15k*\frac{ 2 }{ 3 }}=e ^{10k}\]

Answer 24

thank you very much, its clear