Question

Differentiation help. *question attached below* Will give medal

Answer 1

I am completely clueless as to how to go about doing this mathematical modelling differentiation question. Can I have some help, please?

Answer 2

sorry i have no idea, i've only started differentiating

Answer 3

greatest distance is when x is maximum

x is maximum when dx/dt =0

so first find dx/dt

Answer 4

@hartnn, so do I differentiate x to find dx/dt?

Answer 5

yes

Answer 6

Alrighty, give me a second to work it out

Answer 7

for dt/dt, I've ended up with
\[e ^{-3t}(3-2e ^{t})\]

Answer 8

Oh no, that's not complete.

Answer 9

Step 1: Find derivative
\[\frac{ dx }{ dt }=3(3e ^{-3t}-2e ^{-2t})\]
check out the steps here: http://www.derivative-calculator.net/#

Step 2: When P reaches its greatest distance the derivative is equal to zero
\[\therefore 3(3e ^{-3t}-2e ^{-2t})=0\]
so \[t=\log(\frac{ 3 }{ 2 })+2i \pi n\]
Sorry but I cannot type how I worked out t because its a lot of calculations

Answer 10

3 times that

then equate it to 0

renoto, let not dive into complex and keep the t as log (3/2) only

Answer 11

\(3e ^{-3t}(3-2e ^{t}) = 0\)
solve for t

Answer 12

e^-3t can never be 0

so
3-2e^t = 0

Answer 13

Hmm, my solver is telling me "No solution"

Answer 14

isolate
e^t

Answer 15

then take natural logarithms on both sides

Answer 16

3 = 2e^t

e^t = ..?

Answer 17

3/2

Answer 18

t= ln(3) - ln(2)

Answer 19

yes
thats correct
or its
t = ln(3/2)

now go for next part

Answer 20

differentiate x once again, and you'll get d^2 x/dt^2

then equate it to 0

Answer 21

For the second derivative, I ended up with (12e^-2t) - (27e^-3t)

Answer 22

thats correct
equate it to 0

you may want to factor things out first,
but even if you don't you'll get same final answer :)

Answer 23

(12e^-2t) - (27e^-3t)=0

12e^t = 27

t = ln (27/12) = ln (9/4) = 2 ln (3/2)

see if you get this :)

Answer 24

Yup, got it! :) Thank you so much :D