The volume V(t) of air (in cubic inches) in an adult's lungs t seconds after exhaling is approximately. (Do not use mixed numbers in your answer.)

V(t)= 53+28.5sin(pie(t)/2-pie/2)
(a) Find the maximum and minimum amount of air in the lungs.
Maximum is (cubic inches)
Minimum is (cubic inches)

(b) How often does the person exhale?
every (seconds)

(c) How many breaths per minute does the person take?

sin(pie(t)/2-pie/2) has a max of 1 and a min of -1

so V(t) has a max of 53 + 28.5(1) = 81.5
and a min of 53 + 28.5(-1) = 24.5

b) we need the period
period = 2π/(π/2) = 4

c) easy now

(a) To find the maximum and minimum amount of air in the lungs, we need to determine the maximum and minimum values of the function V(t).

The given function is V(t) = 53 + 28.5sin((πt)/2 - π/2).

To find the maximum value, we need to find the maximum value for the sine function, which is 1. So when (πt)/2 - π/2 = π/2, the sine function reaches its maximum value of 1.

(πt)/2 - π/2 = π/2
Simplifying, we have:
(πt)/2 = π [Adding π/2 to both sides]
t/2 = 1 [Dividing both sides by π]
t = 2 [Multiplying both sides by 2]

So, the maximum value occurs at t = 2 seconds.

To find the minimum value, we need to find the minimum value for the sine function, which is -1. So when (πt)/2 - π/2 = -π/2, the sine function reaches its minimum value of -1.

(πt)/2 - π/2 = -π/2
Simplifying, we have:
(πt)/2 = 0 [Adding π/2 to both sides]
t = 0

So, the minimum value occurs at t = 0 seconds.

Therefore,
Maximum amount of air in the lungs = V(2) = 53 + 28.5sin((π*2)/2 - π/2) = 53 + 28.5sin(π-π/2) = 53 + 28.5sin(π/2) = 53 + 28.5(1) = 53 + 28.5 = 81.5 cubic inches.

Minimum amount of air in the lungs = V(0) = 53 + 28.5sin((π*0)/2 - π/2) = 53 + 28.5sin(0-π/2) = 53 + 28.5sin(-π/2) = 53 + 28.5(-1) = 53 - 28.5 = 24.5 cubic inches.

(b) To find how often the person exhales, we need to find the period of the function. The period of the sine function is given by 2π, which implies that the function repeats itself every 2π units.

In this case, since the argument of the sine function is (πt)/2 - π/2, to find the period, we equate it to 2π:

(πt)/2 - π/2 = 2π
(πt)/2 = 2π + π/2
(πt)/2 = (4π + π/2)
t/2 = 4 + 1/2
t/2 = 9/2

Simplifying, we have:
t = (9/2) * 2
t = 9

Therefore, the person exhales every 9 seconds.

(c) To find the number of breaths per minute, we need to determine how many breaths occur in one minute.

Since the person exhales every 9 seconds, to find the number of breaths in one minute (60 seconds), we divide 60 by 9:

Number of breaths per minute = 60 / 9 = 6.7 (approximately)

Therefore, the person takes approximately 6.7 breaths per minute.

To find the maximum and minimum amount of air in the lungs, we need to determine the maximum and minimum values of the function V(t) = 53 + 28.5sin(π(t)/2 - π/2).

(a) The maximum and minimum values of a sinusoidal function occur when the sine term reaches its maximum and minimum values, which are 1 and -1, respectively.

- The maximum amount of air in the lungs will occur when sin(π(t)/2 - π/2) = 1:
Therefore, we have 53 + 28.5(1) = 53 + 28.5 = 81.5 cubic inches as the maximum amount of air in the lungs.

- The minimum amount of air in the lungs will occur when sin(π(t)/2 - π/2) = -1:
Therefore, we have 53 + 28.5(-1) = 53 - 28.5 = 24.5 cubic inches as the minimum amount of air in the lungs.

So,
The maximum amount of air in the lungs is 81.5 cubic inches.
The minimum amount of air in the lungs is 24.5 cubic inches.

(b) To determine how often the person exhales, we need to find the time period of the function.
In the given function V(t) = 53 + 28.5sin(π(t)/2 - π/2), the coefficient of t inside the sine function is π/2.
Therefore, the time period T is given by T = 2π/(π/2) = 4 seconds.

Hence, the person exhales every 4 seconds.

(c) To calculate the number of breaths per minute, we will convert the time period from seconds to minutes.

1 minute = 60 seconds

Since the person exhales every 4 seconds, we can calculate the number of breaths per minute by dividing 60 seconds by 4 seconds:

Number of breaths per minute = 60/4 = 15 breaths per minute.

Therefore, the person takes 15 breaths per minute.