calculus max/min volume question

Question

The volume of air contained in the lungs of a certain athlete is modeled by the equation $$v=361\sin(91\pi t) +899$$ where $t$ is time in minutes, and $v$ is volume in cubic centimeters.

What is the maximum possible volume of air in the athlete's lungs? Maximum volume= (in cubic centimeters)

What is the minimum possible volume of air in the athlete's lungs? Minimum volume= (in cubic centimeters)

How many breaths does the athlete take per minute?

I am very lost I have no idea on how to approach it any help would be appreciated

Answer 1

I think that you are getting lost in the number of terms in the equation. It's actually a very straightforward question, wrapped in a formula that just makes it look hard.

Ask yourself, what is the minimum and maximum value of $\sin()$ of anything? Since sine itself has a maximum and minimum, it actually doesn't matter what's inside it for min/max calculations, as long as it is continuous. $\sin()$ goes from -1 to 1, period, so it will be at a minimum at -1 and at a maximum at 1. Then calculate the volumes in each case.

Now, on the "breaths per minute", this is based on the inside of the $\sin()$ function.

Answer 2

What is the maximum possible volume of air in the athlete's lungs?

A sine wave's amplitude is increased by the multiplier a in the the equation: $$ y=a*sin(x) $$ This is because a sine wave $y=sin(x)$ has a max amplitude of 1 and you are multiplying it by a.

A sine wave's amplitude is also increased when you add a c at the end like the following equation: $$ y=sin(x)+c $$

This increases the amplitude of the whole sine wave by c, so now the max amplitude of the sine wave is $(1+c)$

Multiplication come before addition in the order of operations, so you end up with the equation $((1*a)+c)$ for the maximum amplitude of:

$$ y=a*sin(x)+c $$

What is the minimum possible volume of air in the athlete's lungs?

A sine wave has a minimum amplitude of -1, so the minimum amplitude of: $$ y=a*sin(x) $$ is $((-1)*a)$

You are also adding to the sine wave like above, so you add c also, for the equation $(((-1)*a)+c)$ as the minimum amplitude of the equation:

How many breaths does the athlete take per minute?

b is the frequency of the sine wave of the following equation: $$ y=sin(b*x) $$ This means that after $(2\pi)$ the sine wave will have gone one cycle (up, down, and back up to it's mid-line). Another way to put this is $(b\ \div (2\pi))$ cycles every time x increases by 1.