calculus max/min volume 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

• Since $361 > 0$, $v$ has its maximum value when $\sin(91\pi t)$ has its maximum value, and $v$ has its minimum value when $\sin(91\pi t)$ has its minimum value. Can you take it from there? Sep 5, 2020 at 0:20
• how would I solve for the maximum volume, what do I plug into equation? im sorry but I have never taken calculus or pre-calculus Sep 5, 2020 at 0:31
• Where does this problem come from? For example, which class is it? Sep 5, 2020 at 0:41
• my college calculus class, he said it's review from pre calculus (I never took pre-calculus in high school). im a college freshman Sep 5, 2020 at 0:43

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.

• now just one more question, where do I plug in the -1 and the 1? would I plug it in as t? Sep 5, 2020 at 4:00
• It's the result of sin(). So, if you just think, sine() is going to be cycling back and forth between -1 and 1. So you can basically ignore t altogether and just realize that the maximum value of the sin() expression will be 1 and the minimum will be -1. Sep 5, 2020 at 19:23

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.

• thank you so much I was so lost this really helps, I will be trying to solve it. again thank you so much I really appreciate it Sep 5, 2020 at 0:58