Maximizing Renewable Energy: Solving for 80% Consumption by 2050

[Phatasmagorick]

Hello everyone.

My name is Zara, I'm new to the forum! I'm in the process of coming up with an applied calculus project as a final project for my calculus I course.

Here are the requirements for the project:

1. Submit one real-world application and their complete solution that requires concepts in Calculus I in order to solve.
2. The real-world application and solution must be original (your own work) and/or cited if other work was adapted. You may not simply copy or modify an existing application and/or solution. Note: Word problems found in textbooks or websites, etc... are not an acceptable applied project.
3. Use APA style to cite any references (websites, books, journals, etc...) that are used to inspire your problems (and/or solutions).
4. The real-world application should be related to current events and should clearly demonstrate concepts in covered in our Calculus I (see Focus on Calculus example).
5. Your grade for this project will be based on (a) originality, (b) accuracy of problem and its solution, (c) relevancy to current events, and (d) relevance and clear connection to concepts addressed in our Calculus I course.

And here is my idea so far (please don't laugh) (Sweating):

In 2013, approximately 13% of the energy sources used in the United States were renewable. According to a report released by GreenPeace U.S.A., it is economically feasible that nearly 80% of U.S. electricity can be produced by renewable energy sources by the year 2050.

a.) If the use of renewable energy was increased by a steady 2% annually from the year 2014, would the consumption of renewable energy reach 80% by the year 2050?

b.) If so, at what year would renewable energy consumption reach 80%?

I need some help on how to devise a formula in order to solve this problem I have created. Also if you have any suggestions on how my questions could be better, I'm open to that as well. I'm not great at this stuff so if anyone could help me find a place to start that'd be awesome.

Thanks for any help.
Zara

 

[Ackbach]

I'm afraid this problem is not complex enough to use calculus, at least it doesn't appear that way to me.

Let $t$ be the number of years since 2013. Let $y(t)$ be the percentage at time $t$ of the US energy resources that are renewable. Then $y(t)=13+2t$. The year 2050 would correspond to $t=37$, so you could plug that into your formula to see whether it exceeds 80 or not.

If I've read your problem correctly, it doesn't use calculus.

 

[MarkFL]

I'm afraid this problem is not complex enough to use calculus, at least it doesn't appear that way to me.

Let $t$ be the number of years since 2013. Let $y(t)$ be the percentage at time $t$ of the US energy resources that are renewable. Then $y(t)=13+2t$. The year 2050 would correspond to $t=37$, so you could plug that into your formula to see whether it exceeds 80 or not.

If I've read your problem correctly, it doesn't use calculus.

I interpreted the problem such that it would lead to the first order IVP:

$$\d{R}{t}=1.02R$$ where $$R(0)=0.13$$

that is if we begin with time $t=0$ in 2013.

Perhaps the OP can state whether a linear or exponential growth curve was intended.

 

[chisigma]

Hello everyone.

My name is Zara, I'm new to the forum! I'm in the process of coming up with an applied calculus project as a final project for my calculus I course.

Here are the requirements for the project:
And here is my idea so far (please don't laugh) (Sweating):

In 2013, approximately 13% of the energy sources used in the United States were renewable. According to a report released by GreenPeace U.S.A., it is economically feasible that nearly 80% of U.S. electricity can be produced by renewable energy sources by the year 2050.

a.) If the use of renewable energy was increased by a steady 2% annually from the year 2014, would the consumption of renewable energy reach 80% by the year 2050?

b.) If so, at what year would renewable energy consumption reach 80%?

I need some help on how to devise a formula in order to solve this problem I have created. Also if you have any suggestions on how my questions could be better, I'm open to that as well. I'm not great at this stuff so if anyone could help me find a place to start that'd be awesome.

Thanks for any help.
Zara

Welcome on MHB Zara!...

I think the best approach to your problem involves the use of discrete mathematics... indicating with $a_{n}$ the absolute value [not in percentage...] of the renewable energy produced in the year n, it is the solution of the difference equation...

$\displaystyle a_{n+1} = \alpha\ a_{n},\ a_{0}=a\ (1)$

... where $\alpha$ is the annual growth rate that for the moment is left undefined... the solution of (1) is very comfortable...

$\displaystyle a_{n} = a\ \alpha^{n}\ (2)$

Using (2) You can find the value of $\alpha$ that satisfies the requirement for the year 2050...

The use of the percentage complicates the problem for two reasons ...

a) for $\alpha> 1$ the solution (2) grows without limits and you are bound not to exceed one hundred percent ...

b) express a percentage You need to know the amount of other sources of energy produced and that makes the problem much more complex ...

Kind regards

$\chi$ $\sigma$

 

[Phatasmagorick]

Thanks so much Ackbach and MarkFL for your responses.

To MarkFL, I was thinking the same thing...that maybe my questions aren't complex enough to need calculus.

And to MarkFL, the way I worded my questions calls for a linear approach I believe. Anyway, I'll keep working on this, but if anyone has any suggestions as far as formulating questions regarding this project that might call for the use of calculus, I'm open to that.

Thanks again guys for the help, I'll try to come up with some better, calculus-based questions relating to global warming and renewable energy usage.

 

[Phatasmagorick]

To Chisigma,

Thanks so much for your response! Thanks to your comments I see how the use of percentage can make the problem more complex.

Your solution is great and again thank you so much. But I have to find a way to use some specific concept that we have learned in my calculus I class and I'm not sure that using discrete mathematics will satisfy that.

Anyway, I'll have to keep working on this but thank you all so much for responding! At the very least you all have shown me that I need to write a better problem and formulate better questions that really call for using calculus concepts.