Weekly Math Problem #80

Equation of a Circle. We're definitely keeping our algebra skills sharp with this week's problem. 🧠 As the saying goes, "Repetition is the mother of all learning". So, you will definitely see skills being repeated in the problems as well the combination of them to solve the problems. The equation of a circle is part of a larger topic called conic sections. Let's  go!  🏃🏿‍♀️

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

       ✔️     Substitution
       ✔️     Completing the square technique
       ✔️     How to solve a quadratic equation

            

WMP #80 wants us to...

Happy solving!

Check back on Saturday, April 9th 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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Part a.

To determine the center and radius of the circle, the completing the square technique needs to be used on the given equation. To start that process, the constant needs to be isolated. Once the equation is in standard form, then the center and radius can be identified.

The center of the circle is (1, -1) and its radius is 1 unit. This was figured out without graphing it. We'll view the graph later in this post.

Part b.

To show that the point (1, -2) is a point of intersection for circle C and the given line, the substitution method was used. The equation of the line was substituted for y in the equation of the circle. The equation of the circle was then simplified into a factorable quadratic equation. This allows for values of x to be solved for...one of which is the x-value in the given point. It is then shown that the corresponding y-value is indeed the y-value in the given point.

Now that we've completed both parts of the problem, let's view the graph:

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

 

▪️ Did you use the same methods above to solve the problem? If so, please share.

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

Thank you for solving with me this week. ✏️
Let's move on to WMP! #81. 💪🏿

Cheers!

The Younge Lady

Weekly Math Problem! #46

Integration. I'll always find my way back to Calculus. 😉 I came across this problem on an exam review sheet. There was a note on the sheet for the students to skip this problem because, it wouldn't appear on the exam. I was intrigued so, it's appearing on the blog this week. 👩🏿‍🏫 I don't really recall solving an integral like this when I first learned integration in college. Since that was too long ago, my recollection is probably off. 

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

        ✔️     How to perform a substitution technique
        ✔️     How to perform completing the square method

(Do you see 👀 the hint I gave above? You're welcome. ☺️) Here is WMP! #46:

Happy solving!

Check back on Friday, April 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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Here is my solution. I hope you can follow it. My goal is for it to be clear. I got started by dealing with the denominator.

Next, I rewrote the original problem with the completed square and saw that I can now perform u-substitution to simplify the integrand.

After completing the u-substitution, I wasn't able to integrate the function. Upon inspection, (hopefully) it can be seen that trigonometric substitution needs to be used to complete the integration.

Ahhh...integration is complete...but the process isn't complete. 🤨 Conversions need to occur so that the answer can in term of x to match the original problem. Plus, I need to figure out what cos 𝜃 is equal to.

Now, the process is complete! 🥳 Let me know what you did differently, or if you didn't do anything differently.

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

Thanks for solving with me this week!

Up next, WMP! #47. 💪🏿

Cheers!

The Younge Lady

Weekly Math Problem! #37

Completing the Square. One thing that can be super annoying--more than solving a problem--is being told how to solve a problem. 😒 Wait a minute. What happened to my mathematical freedom to choose? 😩 It seems like this is done when the instructor has something up his/her sleeve. You may even feel like you are being tricked on purpose. (Sometimes you are.) However, some instructors do that to see if you've payed attention, while some want to see if you have the ability to transfer prior skills to current topics. Even though many instructors have your best learning interests at heart, it is still annoying. 🙄

This week's WMP! is a very good example of the above. To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to solve a quadratic equation
        ✔️     Completing the square method

WMP! #37 says:

Happy solving!

Check back on Friday, February 12th 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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Jumping right in, the first thing that needs to be done is to get the quadratic equation in a form that accommodates the completing the square method to commence--a leading coefficient of 1 and the resulting constant isolated:

Now that the quadratic equation is in the proper form, we can go ahead and employ the completing the square method. (Right now it's incomplete. 🤪) This is where some students get tripped up. Notice that the b-term is not an integer but a fraction. It can be annoying to complete the square when the b-term is a fraction, it can be done. Multiply the fraction by 1/2, then square the result.

So, 4/25 is the number that will allow us to complete the square. Here is how to complete the rest of problem, which involves using the square root method to finish solving for x:

 

Remember, there are two solutions that will satisfy the original quadratic equation we were asked to solve. Lastly, we'll separate the plus-minus and round each solution to the nearest ten-thousandths.

Of course there is a graph. 😁

**This plot was created using Geogebra's Graphing Gaclulator.

▪️ 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!

WMP! #38 here we come!

Cheers!

The Younge Lady