Question

The amount A(t) of a certain item produced in a factory is given by
A(t)= 4000+48(t-3)-4(t-3)^3
Where t is the number of hours of production since the beginning of the workday at 8 am. At what time is the rate of production increasing most rapidly?

Answer 1

find double derivative and set equal to zero
solve

Answer 2

and make sure the first derivative is positive

Answer 3

48-12(t-3)^2 is the first dirivative?

Answer 4

looks right to me

Answer 5

when i got it equal to 0 i got t=
3

Answer 6

from there i dont know

Answer 7

you set the second derivative to zero
-24(t - 3) = 0
t = 3

Answer 8

yes, then what?

Answer 9

it reaches the most rapid in 3 hours
you start at 8 am
plus 3 equal 11 am

Answer 10

yo man you just blew my mind!
thank you sooo much! now i know but can you explain a bit? why the 2nd derivative?

Answer 11

sure
the first derivative will give you the slope yes?
if the slope is positive, then the rate at which it's being produced is increasing
now here's the confusing part, so bare with me
the second derivative will give you the slopes of the rate of production
just like how when the first derivative hits zero, it's a max/min
when the second derivative hits zero, it's when the rate of production is max/min
so you solve for that

make sense?

Answer 12

i have to go now
but if this doesn't make sense, i'll try coming on again later
good luck!

Answer 13

no it made perfecet sense! I just dont pay attention in class as much, ha! well ok thank you and have a good day!