Weekly Math Problem! #41

Arc Length. Nothing strange going on here this week. I'm just getting back to a calculus problem I wanted to do last week, this week. A little application situation in the form of finding the arc length. 👀 There are times when it is needed to find the length of a curve and there aren't any regular geometric techniques that'll help with that, so calculus steps in to save the day. 😁

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to find the first derivative
        ✔️     How to work with hyperbolic functions
        ✔️     How to do integration

WMP! #41 says to:

Happy solving!

Check back on Friday, March 12th for the solution, which will be posted below ⬇️.

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Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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And we're back to get this problem done. So, let's see... The problem is asking us to find the arc length of a function on a closed interval. There is a formula for such a situation.

In regards, to our problem, we need to identify the function, take its first derivative, and use the interval. Starting with the function, we're dealing with hyperbolic cosine who has a first derivative that is hyperbolic sine.

Now that we have the first derivative of the function and the interval, we can plug the information in the formula and perform the integration. Remember, we will come out with a numerical answer that represents the length of the function (a curve) for the specified endpoints; it makes that we are working on a definite integral.

As you can see, the arc length of f(x) = cosh(x) on [0, ln 2] is 3/4 units. Let me point out that a scientific or graphing calculator can easily evaluate hyperbolic functions for you. However, I still evaluated it by hand so you can see exactly where the final answer comes from. To assist in evaluating by hand, the exponential representation of hyperbolic cosine is used.

Here is an image of the given function and the part of the function for which we found the arc length.

 

**This plot was created using Geogebra's Graphing Calculator. Click image to enlarge.

 

▪️ We're you able to find the arc length?

▪️ Leave your response down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Moving right along to WMP! #42. 👩🏿‍🏫

Cheers!

The Younge Lady

Weekly Math Problem! #36

Exponential Application. Math makes the most sense to me when it has a real-world application. (Oh how is wish theory made more sense to me.) So, every now and again, I like to get word problems in. (**Whispering** But that doesn't mean I love word problems. I still find them scary sometimes.) 

This week's problem is very relatable to everyday life. What is the likelihood that you (as the driver) will get into an accident with a certain blood alcohol level? It makes sense that the more intoxicated you are, the less alert you are, making it easier for you to get into an accident. (That's the best way I could say it, although there is probably a more accurate way to express this.) To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to solve an exponential equation
        ✔️     How to isolate the exponent in an exponential equation

WMP! #36 says:

See problem here.

Happy solving!

Check back on Friday, February 5th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Let me just start with where I went wrong in solving this problem. 😕 Here is what I did initially:

Can you see what the problem is? It isn't the steps I used that are incorrect. It's the answer I got. When I saw that the answer was negative, I felt like something was wrong. So, the first thing I did as go back over my solution to see if there was an error in the computation. I didn't find anything. Next, I decided to plot the function to see what the curve looked like. Hmm... 🤔 The curve looked fine to me but, when I plotted the risk from the problem with the corresponding blood alcohol concentration (BAC) I solved for, it further confirmed that a negative answer wasn't it. I quickly looked up BAC numbers and was informed that in the US, they should be positive ranging from about 0.08% (0.0008) to 0.40% (0.0040). Okay.

Then, I went back to the word problem and read it again to see what it said. (...And this is why word problems aren't my friends.) The problem said "R is a percent." Noticed I let R = 0.25. Why? I am so used to converting a percent to its decimal representation when working with exponential equations, that I was so sure R should be 0.25 instead of 25. 🙄 I read that thing and didn't even think that R should be a whole number. 🤷🏿‍♀️ Here is my corrected calculation:

According to this model, the BAC that corresponds to a 25% accident risk is approximately 0.11% (0.0011). *The problem didn't say what the unit for BAC is. Now that I am more familiar with it, I know not to convert the 0.11 to 11%.*

This problem definitely had me do more work that I thought I would. No worries. I won't stop solving...and neither should you. 

▪️ How much did you know about BAC?

▪️ How did you find this week's problem?

▪️ Leave your response down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Onward to WMP! #37

Cheers!

The Younge Lady

Weekly Math Problem! #21

Integration. So, here we are again with another integration problem. If you've done last week's problem, then you've already done some of the work needed to do this week's problem. 🤓 Wait, you haven't seen last week's problem?! Pause....check out WMP! #20. 

Last week I mentioned that partial fraction decomposition is an important technique that allows you to "simplify" some complicated rational expressions by breaking them down into smaller parts. WMP! #21 is such a problem that requires the technique of partial fraction decomposition.

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️ How to integrate rational expressions

        ✔️ Partial fraction decomposition

WMP! #21 wants us to...

Happy solving!

Check back on Friday, September 25th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Again, if you take a look at WMP! #20, you will see that it has the same rational expression as this week's problem. So, when you're given a rational expression, similar to that one, to integrate, you can see what needs to be done to carry out the integration. Here's how I carried it out:

That's all I've got! (I actually really like this problem.)  

 

▪️ How did you feel about this week's problem?? 

▪️ Did you carry out the integration differently than I did??

▪️ Comment below with your responses and let me know what you thought about this week's problem.

See you right back here next week for WMP! #22, the last problem of September.

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Cheers!

The Younge Lady

Weekly Math Problem! #6

Quadratic Function Application. One of the things that people cannot stand in mathematics, next to fractions is...dare I say it...⚠️word problems⚠️. 🙄 Trust me I get it! 😒 I've encountered hundreds, if not a few thousand, words problems over the years I've been doing mathematics. I'm talking about word problems from algebra, geometry, precalc, calc, differential equations, linear algebra, statistics, standardized tests such as the SAT and GRE, not to mention word problems from the sciences. And still, when I see a word problem, my insides jump just a little bit. It can definitely be tricky to relate the words you're seeing in a problem to some math formula/techniques that you'll need to solve the word problem. If you're super annoyed with word problems, leave a comment down below 👇🏿 and vent a bit.

To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  How to convert words/phrases to algebraic expressions

    ✔️  How to solve a quadratic equation

    ✔️  How to find the minimum/maximum value of a quadratic function

You will also need to know a little finance vocabulary, which can be easily looked up. Here goes WMP! #6... 

Happy solving! 

Check back on Friday, June 12th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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One of the things that will be helpful when solving a problem is identifying and marking the words/phrases that can be translated into algebraic expressions/equations. In the solution, the key phrases are denoted in blue. 

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〰️〰️〰️〰️〰️

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Below is the graph of R(x): 

*This graph was generated on Desmos.com.*

The graph of the function isn't part of the original question, however it's great if you can have a visual to confirm or deny the answers you came up with. I showed the point on the graph where no revenue is generated, which is one of the roots (or "zeros") of the function. I also showed the point on the graph where maximum revenue is generated, which is the maximum point (or turning point).

◾️ How did you find this word problem?? Easy? Hard? In between? 

◾️ Comment below with your responses and let me know what you thought about this week's problem.

See you again, soon, for WMP! #7 🤓

Cheers!

The Younge Lady