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! #35

Polynomial Division. Alright folks. I think I've mentioned this before but, I'm not sure. So, I am mentioning it now. I do my best to make sure I am mixing up the type of problems I post every week. I have learned different types and levels of math over the years and have liked most of it. (Although, I possess a bias for calculus.) As a tutor, I want to make sure that I can help others with various levels of math, especially at the college level. One of the things I have to consider when choosing a problem is making sure that I am able to type the question and it's solution. This week's problem is good. 👍🏿 (I don't know if that's a hint or not. 🤷🏿‍♀️)

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

        ✔️     How to divide a polynomial by a binomial

Ready? Great! WMP! #35 says:

Happy solving!

Check back on Friday, January 29th 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 numerator in the problem (which is also the dividend) is prime--it cannot be factored. Since, it cannot be factored and reduced, this division needs to be completed the long way...literally ---> via long division. 🥴 Sorry...

Unfortunately, I couldn't show each individual step of the long division process. (A video is needed for that.) What I've done, instead, is just showed the completed long division in two forms--the first with the undistributed negative sign and the second with the distributed negative sign. **Notice that the dividend lacks a cubic term. So, to proceed with the long division method, a place holder must be used:  0x3 

 

    

As you can see there is a reminder. The reminder should be written as a fraction, with the divisor as the denominator. Putting that along with the quotient shows the final answer as follows: 

Wait...not so fast. Let's check it! (That's one of the things that I like about math--you can check your work). To check, we'll multiply the quotient and the divisor, adding the reminder to it.

Now, you're free to go. The check works out and I am satisfied. 

▪️ Did your solution work out?

▪️ 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! #36

Cheers!

The Younge Lady

It's My First Blogiversary!

WOW! A year ago, today, I published my first blog post. Wait...what's that? You didn't read it? 😲 Let me help you find it. Who's That Lady?! was published on Monday, January 20, 2020. ☺️ When I created this blog, I didn't really know how it was going to turn out but, it just felt right to do. After publishing my introductory post, my first official WMP! post wouldn't be published until a few months later in May of 2020. I was determined not to let the first week in May pass by without getting the ball rolling on these problems. I'm so glad I did. 🙌🏿

I wanted to do something a little different to commemorate this occasion, so I created a word search puzzle with some words associated with my blog. Wanna play?

 First Blogiversary Puzzle! PDF

Happy finding!

I appreciate your support--solving with me or just clicking the link to see what problems I posted. I will definitely keep posting problems and am hoping to sprinkle in other types of posts. Maybe I can start doing some videos this year. 🤷🏿‍♀️ We'll see. What do you think? Share your thoughts below ⬇️.

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

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