Weekly Math Problem #62

Improper Integral. Ever so often, I find my way back to calculus, and this week it's via an improper integral. By my recollection, the technique is okay. The type of function you're dealing with can definitely be the cause that the problem is annoying to do. Let's just hope this week's problem isn't annoying to you. 😬 

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

        ✔️     How to solve an improper integral
        ✔️     Methods of integration

     

WMP! #62 says to...

Happy solving!

Check back on Saturday, November 6th for the solution, which will be posted below ⬇️.

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✏️📓 Solution Time! 📓✏️

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Alright...let's get into this problem! If you looked up integral tables and used a calculator, that's good. I did too! I wasn't expecting anyone to have the information needed to solve this problem memorized. I didn't have it memorized, but I felt confident that I knew how to use the assistance.

I started this problem by using an integral table. Why? Well, I saw that the function was rational and knew that most, if not all, integral tables have some rational functions on them. When I saw what I needed, I rewrote the function in a way that help me begin solving the problem.

With the problem rewritten, I could see that I needed to invoke the substitution method to keep going. Since the substitution method uses a new variable, I also adjusted the lower bound of integration to match the new variable.

Once that was done, I could now move forward and integrate.

Since this is a definite integral, we need to evaluate! Evaluating the second term, which comes from the lower bound was simple. If you're familiar with this special value, then you can simple write the value. You can also use a calculator like I did. 😁 It's the first term, that requires a bit more work. Infinity isn't a value to plug in, so I used my calculator, again, to see if the function converges as my variable approaches infinity. It does!!!

Now that I have the values I need, I can plug them in and simplify.

I don't want to leave my answer like that, so I did a little "clean-up" by rationalizing the denominator.

Here is an image of the function. Keep in mind that the variable is approaching infinity. Even though, the shape doesn't end, the area under the curve converges to value above. 

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

▪️ Were you able to solve the improper integral??

▪️ Let me know what you thought about this week's problem in the comments section. 

Thanks for solving with me this week!

WMP! #63. is up next. 👩🏿‍🏫

Cheers!

The Younge Lady

Weekly Math Problem #61

A Hybrid Equation. Yeah...I don't know what else to call it. 🤷🏿‍♀️ I was going through an old math worksheet for the NYS Algebra 2 Regents Examination from JMAP, when I stumbled across what is now this week's problem. 🤔 I don't recall solving anything like this in the past...but I could definitely be wrong. If I have, then it was a very long time ago. This is why I chose the problem. Based on the math I know, I am attempting it this week! Why don't you join me?

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

        ✔️     How to solve equations involving absolute value expressions
        ✔️     How to solve equations involving quadratic expressions

     

WMP! #61 want us to...

Happy solving!

Check back on Saturday, October 30th 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. 😆 )

Weekly Math Problem #60

Logarithmic Equation. Honestly, I don't want to lose momentum this week. I want to say more but...I...am...tired! 😩😩

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

        ✔️     Properties of Logarithms
        ✔️     Methods to solve a quadratic equation

     

WMP! #60 want us to...

Happy solving!

Check back on Saturday, October 23rd 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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If you've solved the problem above, then you would've seen that, indeed, solving a quadratic equation was required. There are options to solving a quadratic equation. How many options you have available to you depends on the type of quadratic equation you have. Below, you will see that I chose to use the completing the square method to solve the resulting quadratic equation. 

Did you forget that I like visuals? (I don't blame you if you did. 😅) The graph below shows the intersection where the LHS of the equation meets the RHS of the equation. As you can see, the intersection occurs when x=2...our solution! 👏🏿

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

▪️ Were you able to solve this week's equation??

▪️ Let me know what you thought about this week's problem in the comments section. 

Thanks for solving with me this week!

Up next...WMP! #61. 👍🏿

Cheers!

The Younge Lady

Weekly Math Problem #59

Venn Diagram. When I think about it, I learned about a Venn diagram decades ago without really knowing anything about Venn. I used to think, "What is a Venn?", instead of thinking, "Who is Venn?" 🤷🏿 I can't recall ever looking up Venn, so it looks like I am doing this for the first time. Venn is John Venn. John Venn was an Englishman born in 1834 who became a mathematician and philosopher. (This information was pulled from Wikipedia : John Venn.) John Venn was also part of the #beardgang. 😁 Anyhow, I used to find Venn diagrams confusing, when first learning them. Now, I definitely get it. 👍🏿

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

        ✔️     How to understand a Venn diagram
        ✔️     Simple probability

     

Ready or not, here is WMP! #59:

Happy solving!

Check back on Saturday, October 16th 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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I don't know how you feel about Venn diagrams, but I definitely understand why working with them can be confusing at times. I started this problem by identifying the cardinality of each set. There are three sets represented in this problem--R, J, and Zeta (the universal set). The cardinality of a set is the amount of elements in the set. In this problem, algebraic expressions are used to represent some of the cardinalities. "n(set_name)" is the notation used for cardinality.

Before I can move forward to find the probability of interest, I need to know what the exact cardinality of set R is. This requires me to solve for x. I can use both representations for the cardinality of set Zeta to do this.

Here are the exact cardinalities for the three sets:

Now, I have the information I need to find the probability of interest.

The probability that a randomly chosen car is 7/30.

▪️ Did you find the probability??

▪️ Let me know what you thought about this week's problem in the comments section. 

Thanks for solving with me this week!

Up next...WMP! #61. 👍🏿

Cheers!

The Younge Lady

Weekly Math Problem #58

Complex Numbers. Lots of people don't find it easy to work with real numbers (and I don't blame them). We're talking all of the major operations...plus others. Then finding out that there are "fake" or imaginary numbers can be very off-putting! 😨 (They're complex for a reason.) Nonetheless, they exist and we can do the same things with them that can be done with real numbers. Division is one of those operations.

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

        ✔️     Rationalizing the denominator
        ✔️     Understanding complex conjugates

     

WMP! #58 wants us to...

Happy solving!

Check back on Friday, October 8th 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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This week's problem asked us to divide complex numbers, specifically divide conjugates. This requires us to rationalize the denominator. Remember, the i is concealing an irrational number. Below is my solution for this week's problem:

▪️ Did you get the same answer??

▪️ Let me know what you thought about this week's problem in the comments section. 

Thanks for solving with me this week!

Up next...WMP! #59. 👍🏿

Cheers!

The Younge Lady