Question

A normal adult breathes in and exhales about 0.84 litres of air every 4 seconds with the minimal amount in the lungs of 0.08 Litres. Write a COSINE equation with 0≤t≤8 and find the time of maximum air capacity in this interval.

Answer 1

Suppose the total capacity of lungs be V0
Now, let y be the amount of air in lungs at any moment t
Then, y(t)= a + (b* cos(w*t)) here a,b,w are unknown constants

We need to determine their value from the given information; the minimum amount in lungs= 0.08 the minimum of the equation on board is a-b

hence, we get to write, a-b = 0.08

Agree thus far? :)

Answer 2

How do you know if a - b equals the minimum amount.

a = amplitude
b= horizontal compression or expansion of the graph.

Answer 3

a= constant amount

b= amplitude

when y obtains minimum value, cos(wt)= -1
so, y becomes a-b

Answer 4

No, please write it in this form if it is possible.

y= a cos(b (x -c )) +d

a = amplitude

b= horizontal expansion or compression

c = phase shift

d = vertical shift or translation

It is easier for me to understand what you tell me.

Answer 5

I agree to your minimum value = 0.08 L

\[\left| \min \right| = d + a\]

Answer 6

yes

my bad 0.08 = d +a

Answer 7

b = 2pi /4 or pi / 2

Answer 8

Right. The first thing to note is that there is no phase shift we are considering here, so c=0 now, min is 0.08 = d -a min is NOT d+a d+ a will be the max

now, as I said earlier, it takes 4 seconds to repeat the cycle. hence T= 4 in our form, x stands for time t now, T=4= 2pi/b

Answer 9

so we get b= 2pi/4 = pi/2
A normal adult breathes in and exhales about 0.84 litres of air every 4 seconds
b= pi/2

now, we need to write for 0.84
0.84 is the difference between the max and the min. The min value was already found to be d -a ... the max value will be d+a

Answer 10

the difference between max and min is (a+d) - (d-a)
which turns out to be 2a and that equals 0.84
2a= 0.84 , agree?

Answer 11

perfect :)

Answer 12

Alright, now solve for a,d
we got 2a=0.84
we got d-a = 0.08

so, what are a, d?

Answer 13

a= 0.42

d= 0.42 + 0.08 = 0.50

Answer 14

Right! :) and we already found c=0
b=pi/2
so, can you now write the equation?

Answer 15

y = 0. 42 cos [ pi/2 (t)] + 0.50

Answer 16

Exactly, you got it! x'D

Answer 17

find the time of maximum air capacity in this interval.

Answer 18

Is it 0 and 4?

Answer 19

Oh... that happens when cos(bt)=1
So, t=0 is definitely a choice and t=4 is also a choice :)

Answer 20

since 0≤t≤8 t=8 is also a choice

Answer 21

so t=0,4,8

Answer 22

but I have a very weird answer of it from my teacher which does not make sense

I will write it in a sec

Answer 23

Maximum capacity is when cos (pi . t /2)= -1

pi . t/2 = pi --> t =2

pi .t /2 = 3 pi --> t =6

Air capacity is maximum at 2 sec and 6 sec.

Answer 24

I do not understand that logic. I would think that t=2 and t=6 corresponds to minimum air capacity.

Answer 25

I think you should ask this to your teacher...

Answer 26

OK, but we managed to go through most of it. I appreciate your help :)

Answer 27

:)

Answer 28

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