# Function combinations

#### otownsend

Hi,

I hope someone can help. I'm trying to develop the equation for the following word problem:

This is a question from my pre-calculus course, where we are expected to use our knowledge of function combinations to derive an equation for this context. However, I'm finding it to be rather difficult to derive an equation without any knowledge of physics that is at minimum a grade 12 level - which I unfortunately do not have.

Here are the characteristics of the function that I'm pretty confident of:
- the equation will be a product of an exponential and cosine function
- the amplitude is 20
- the equation of axis is 30
- for some reason, I found somewhere saying that the period is pi/2, but I don't fully understand why this would be the case

If there is a simple explanation to how this equation could be derived that doesn't involve too much physics and advanced functions-level math, please let me know... I'm quite stuck!

#### skeeter

MHF Helper

with no friction ...

$d = 20\cos(\pi t) + 30$

... where $d$ is in cm and $t$ is in seconds.

with friction, two sets of coordinates $(t,d)$ ...

$(0,50)$ and $(10,33)$ $\implies d = f(t) \cdot \cos(\pi t) + 30$

At $t=0$ and $t=10$, $\cos(\pi t) = 1 \implies f(0)=20$ and $f(10)=3$

since $f$ is exponential ...

$f(t)=x_0 \cdot e^{kt}$

$f(0)=20 \implies x_0 = 20$

$f(10)=3=20e^{k \cdot 10} \implies k= \dfrac{\ln(3/20)}{10} \approx -0.19$

therefore ...

$d(t) = 20e^{-0.19 t} \cdot \cos(\pi t) + 30$

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#### otownsend

Awesome thank you for your help. That makes a lot of sense.