doesnt make sense?

[Sanguinne]

there is this question in my textbook i cant seem to solve

The length of a snake, L cm, at time t weeks after it is born is modelled as: L=12+6t+2sin((t x pi)/4), 0≤t≤20

Find
a) the length at i) birth and ii) 20 weeks
b) R, the rate of growth, at any time,t
c) the maximum and minimum growth rat

ive managed to do a and b but i cant solve c. The way i attempted to solve it was let the rate of growth which is 6+Pi/2cos((pi x t)/4) and let it equal 0. when i try to work it out i end up with

(pi x t)/4 = cos^-1(-12/pi)

the answer in the book for c is

max = pi/2 + 6, min = 6 - pi/2

[abcdqd]

cos ((pi*t)/4) can take values between -1 and 1, so the maximum will occur when cos ((pi*t)/4)=1 and the minimum will occur when cos ((pi*t)/4)=-1

[Phy124]

Think about the minimum and maximum of any sinusoidal function. The maximum will be the vertical translation plus the amplitude and the minimum will be the vertical translation minus the amplitude. The reason for this being the case was explained by abcdqd.

The way i attempted to solve it was let the rate of growth which is 6+Pi/2cos((pi x t)/4) and let it equal 0

By doing this you are finding when the growth rate is equal to zero not when it is at a minimum/maximum. I assume you were thinking of equating the derivative to zero (and consequently subbing in the values of t back into the growth rate equation to find the minimum/maximums), which indeed can be done. However, when it comes to sinusoidal functions this is pointless when the answer is right in front of you (see first paragraph)

[Sanguinne]

essentially, if i wanted the maximum i would let cos(pi*t/4)=1 and if i wanted a minimum i would let it equal to -1

[Phy124]

essentially, if i wanted the maximum i would let cos(pi*t/4)=1 and if i wanted a minimum i would let it equal to -1

If you were looking for the t value at which the minimum/maximum occurred, rather than the actual values of the min/max, yes.

[Sanguinne]

would i also do this for graphs which are sin and tan?

[silverpixeli]

not tan, because the tan function is NOT restricted to values between -1 and 1
but sine, yes.
so say you have a sinusoidal function that is something like this: 13+3sin(3t-4pi) and i'm just making that up out of thin air
if you want the maximum, then the largest possible output for sin(3t-4pi) is 1, which means you replace sin(3t-4pi) with 1 in your formula and it becomes:
13+3(1)
=13+3
=16

similarly, if you want the minimum, let the function=-1 because that is the smallest output
13+3(-1)
=13-3
=10

Note, if the rule was instead 13-3sin(blah) then the maximums and minimums would be reversed, because when sin(blah) is -1 it ends up being positive and vise versa

if you want the value of t that will give you the max/min, you let the function =+-1 and solve for t

eg what value of t gives a maximum in 13+3sin(3t-4)
for it to be a maximum, sin(3t-4) must be 1
sin(3t-4)=1
solve for t (you'd normally have to consider a restricted domain)

for minimum, same deal applies, except you solve for sin(blah)=-1

same works for cosine, but NOT tangent
also, if the rest of the function is not a constant, this wont work [eg y=3x+sin(x)]

[Sanguinne]

great explanation

thank you very much