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! #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 ⬇️.

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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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! #34

Inverse of a Function. I have different ways of choosing problems to solve on this blog. Sometimes I go through books and notes I have. Other times, I use Google. 😁 This week, Google won as my method due to a lack of time. 🤷🏿‍♀️ What can I say? So, I decided to check out some precal topics and found a lighter problem to use. 

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

        ✔️     How to find the inverse of a function

That's it? Yup...that's it. WMP! #34 wants us to...

Happy solving!

Check back on Friday, January 22nd 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's go right on ahead and jump in this thing! 

Here comes the fun 🤪 part! (At least I think it's fun. 🤷🏿 ) The equation right above here is a literal equation. We are going to solve it but, the answer we get will not be a number. By placing x over 1, it can be more clearly seen that the above equation is a proportion.

We're just about done. Just one more thing...a mere formality:

That's it! We've found the inverse function, f -1. 🤸🏿‍♀️ Yay! Like the original function, the inverse is also a rational function, with a variable present in the denominator. That means we cannot use, or plug-in, any value for x that we like. To continue with the second part of the question, we will find the domain for f -1 by identifying which value of x will cause the function to be undefined. I did something slightly unconventional to find the value. Instead of setting the denominator equal to zero, I set the denominator not equal zero and proceeded normally:

Now, I can write the domain formally. I don't really have a preference for notation, so I did it two ways:

 

You know I wouldn't end this blog post without graphing the functions, right? Okay, great! (I tried something a little different with the color scheme. What do you think?) When a function and its inverse are plotted on the same set of axes, they will be mirror images of each other. The mirror is the line y = x.

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

▪️ Let me know what you thought about this week's problem down below.

Thanks for solving with me this week!

Up next...WMP! #35

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Cheers!

The Younge Lady

Weekly Math Problem! #33

Equation of a Tangent Line. And I'm back with the first WMP! of 2021. New year, same goal--to solve various types of math problems regularly so that the ol' noggin 🧠 stays sharp! 💪🏿 Do you have any math goals for 2021? Drop them below ⬇️.

Since calculus is a fave, I thought I'd start this year with a calculus problem. (It'll help me with momentum). I don't recall if I have mentioned this before but, my first experience with calculus was in high school--12th grade. I was preparing for the AP exam so, we were provided with materials to aid us with preparation. Here is a picture of one of the books I used in high school and where this week's problem is coming from:

I was enough of a nerd 🤓 then hold on to the book for almost 20 years 👵🏿. 

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

        ✔️     How to find a derivative
        ✔️     How to find the equation of a line using given information

WMP! #33 wants us to...

Happy solving!

Check back on Friday, January 15th 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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Just in case you're not familiar with type of equation we've been given in the problem, it is the equation of a circle, centered at the origin, with a radius of 13 units. We will use the method of implicit differentiation to find the derivative:

Now that we have a point and a slope, we can proceed to finding the equation of the tangent line: 

You can stop at this point, but let's take a few extra steps to write the equation in a "nicer" form:

We're done. Now, if you remember anything about from last year, you will recall that l like images. So, I would be remiss if I didn't add to this blog post an image of the circle and it tangent line at the point (5, -12):

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

...And that's a wrap! The first WMP! of 2021 is complete. I am excited to keep going this year and see what what other types problems we can solve.

▪️ What kind of problems would you like to see this year??

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

Thanks for solving with me this week!

Can you see it? WMP! #34 is next.

👁👁➡️

Cheers!

The Younge Lady

Weekly Math Problem! #28

Two-Variable Nonlinear System. One of my goals with this blog is to challenge myself to solve/do problems that I'm not used to. Over the years, I've solved many linear systems of equation using a the three methods--graphing, substitution, and elimination. I haven't solved nearly as many nonlinear systems, especially ones that look like the one we're solving this week. However, after examining the problem, I know how I want to algebraically tackle this question. C'mon and tackle with me...

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

        ✔️     Substitution method
        ✔️     Elimination method
        ✔️     How to solve a proportion
        ✔️     How to solve a quadratic equation

WMP! #28 wants us to...

Happy solving!

Check back on Friday, November 20th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady

✏️📓 Solution Time! 📓✏️

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As mentioned before, I knew how I wanted to algebraically tackle solving this system after examining it for a bit. Okay, okay....who am I kidding? I kind of did both. 😊 L👀k...

So, the solution to this system has four points for the solution--{(1, 1/3), (1, -1/3), (-1, 1/3), (-1, -1/3)}. However, I was still curious to see what the substitution method looked like. There is more than one way to apply the substitution method, so here's how I did it... 

As you can see, solving for y using the substitution method the way I did, does yield the same values for y I got with the elimination method. Whew! If I continued to solve for x with the substitution method, I will also get the same values as I did above. 

Now, you know I couldn't end this problem without having a graph of the system. The graphs have been labeled with an "A" and "B" in accordance with how I labeled them above.  

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

▪️ What method did you use to solve this nonlinear system??

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

Thanks for solving with me this week!

Up next...WMP! #29

💪🏿➡️

Cheers!

The Younge Lady

Weekly Math Problem! #15

Quadrilaterals. Listen, geometry is not necessarily a favorite topic in mathematics for a lot of people. So, now you're thinking, "What about you, Lori? How do you feel about geometry?"  Well, it's not my favorite either 😅; I'm not in love with geometry. 🤷🏿 I will say, I am very visual, so I love the fact images, and the need to draw pictures, are a crucial part of geometry. I have lots of respect for this topic. Geometry is everywhere.

This week's problem was taken from the last administered NYS Regents High School Examination in Geometry before quarantine--January 2020, Part III, question 32. To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  Plotting points on the coordinate plane

    ✔️  Slope, distance, midpoint formulas

    ✔️  Properties and theorems for quadrilaterals

    

WMP! #15 says...

Happy solving! 

Check back on Friday, August 14th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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In high school I learned the two-column proof method for geometry proofs. The proofs I remember doing the most were triangle proofs. I rarely, if ever, used the paragraph-style for writing geometric proofs. So, for this problem I'll complete the necessary computations and accompany them with statements that support those computations. 

There are various way to prove that a quadrilateral is a rhombus. This is because there are various ways to prove that a quadrilateral is a parallelogram, of which a rhombus is a special case. I will prove that quadrilateral NATS is a rhombus with the following method:

            I.   Prove that quadrilateral NATS is a parallelogram

            II.  Prove that the parallelogram is a rhombus.

 

But first, here is an image of quadrilateral NATS (the rhombus in question):

**This plot was created using GeoGebra's Geometry App.

I. Prove that quadrilateral NATS is a parallelogram. I will do this by showing that the diagonals of NATS bisect each other. The midpoint formula was used.

Diagonal NT and diagonal AS share a midpoint. NM = TM and AM = SM. Therefore, diagonals NT and AS bisect each other. Therefore, quadrilateral NATS is a parallelogram. ✔️

II. Prove that parallelogram NATS is a rhombus. I will do this by showing that the diagonals are perpendicular. 

The slopes of diagonal NT and diagonal AS are negative reciprocals of one another. Therefore, diagonal NT and diagonal AS are perpendicular. Therefore, parallelogram NATS is a rhombus. ∎

It's been quite a while since I've done one of these. This has me thinking 🤔 back to when I took my high school math Regents. I remember doing logic proofs. Maybe I'll explore that topic in a future WMP!

◾️ How do you feel about geometry?? 

◾️ What method did you use to complete the proof?? 

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

Without further ado, on to WMP! #16 

Cheers!

The Younge Lady