Exponential functions (calculator exercise)

[Joshuaniktas]

Hi there, I have tried to do these questions but I don't understand. Any help would be appreciated!

 

[MarkFL]

Hello, and welcome to MHB! :)

Let's begin with part (a). We are to find when:

\(\displaystyle f(t)=g(t)\)

Or:

\(\displaystyle e^{-\frac{t}{20}}+10=te^{-\frac{t}{20}}+6\)

Let's subtract 6 from both sides:

\(\displaystyle e^{-\frac{t}{20}}+4=te^{-\frac{t}{20}}\)

And then arrange as:

\(\displaystyle 4=te^{-\frac{t}{20}}-e^{-\frac{t}{20}}\)

[DESMOS]{"version":7,"graph":{"viewport":{"xmin":-34.31311077614891,"ymin":-11.279300231558665,"xmax":104.86022436822739,"ymax":62.76481974796097}},"randomSeed":"a115e9332f088699d9a4c7866dadfcce","expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"y=4e^{\\frac{t}{20}}"},{"type":"expression","id":"2","color":"#2d70b3","latex":"y=t-1"}]}}[/DESMOS]

As we cannot solve this algebraically, we will need to rely on a CAS to generate numeric approximations:

https://www.wolframalpha.com/input/?i=4e^(t/20)=t-1
We get:

\(\displaystyle t\approx6.54997\)

\(\displaystyle t\approx50.1865\)

Now, for part (b), let's look at this graph:

[DESMOS]{"version":7,"graph":{"viewport":{"xmin":-9.635503167985057,"ymin":-15.7165503991205,"xmax":71.01286536037324,"ymax":27.190644928716516}},"randomSeed":"5093a3ded45d142de4030d436ace2162","expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"y=xe^{-\\frac{x}{20}}+6\\left\\{0\\le x\\le60\\right\\}"}]}}[/DESMOS]

Click on the function's definition on the left to make it active, and you will see the maximum point on which you can click...what do you see when the point is labeled?