Question

Can someone help with a couple optimization problems. Calculus AP.

Answer 1

I have finished the first 5.

Answer 2

@mathmate

Answer 3

@agent0smith if you got any time could you help please?

Answer 4

That's the easy part. Find the max on the graph (you can see it on the graph i posted in the last one)

Then differentiate the function to find the max.

Answer 5

okay

Answer 6

this graph

Answer 7

Yep and you can find the derivative to find the max exactly.

Answer 8

oh so find the derivavtive first or the maximum first?

Answer 9

What do you mean?

Answer 10

Do i find the derivative of A(t) then find the max of that or find the max of just A(t)?

Answer 11

Well to find the max, you need the derivative...

But the question tells you to estimate from the graph.

Answer 12

oh okay i would estimate it to be about 80,000

Answer 13

the max of A(t) is 2500e^(413/120)

Answer 14

and it is at x = 625/36

Answer 15

Did you get it with using the derivative though?

Answer 16

no i used wolfram

Answer 17

Well you'll have to show work :P

Answer 18

yeah your right okay so now i need the derivative?

Answer 19

\[\Large A(t)= 2500 e^{0.5 \sqrt t+ 0.06(40-t)}\]just the fact that the derivative of e^f(x) is f '(x)e^f(x)

Answer 20

f '(x) in this case would be \[0.5\frac{ 0.5 }{ \sqrt{x} }-0.06\]

Answer 21

@agent0smith

Answer 22

Keep going... see my last post.

Answer 23

oh okay

Answer 24

\[2500e ^{0.5\sqrt{t}+0.06(40-t)}(0.5\frac{ 0.5 }{ \sqrt{t} }-0.06)\]

Answer 25

And you know you have to set it equal to zero to find the max.

Answer 26

Okay

Answer 27

this will take awhile

Answer 28

No, it won't. You should also know that e^f(x) can never equal to zero.

Answer 29

Which only leaves one factor that can be equal to zero.

Answer 30

f '(x)

Answer 31

@agent0smith i have concluded that the maximum value for this is 77,967.40

Answer 32

Seems reasonable.

Answer 33

So how does this relate to when he has to sell the card as asked to find

Answer 34

Well you were meant to find the t value, so what was it?

Answer 35

17.36

Answer 36

Looks right

Answer 37

alright cool thnx mate!!