Weekly Math Problem #68

Binomial Distribution. I just thought I'd end this year with a "lighter" problem. This week's problem, which by the way, is the last WMP for 2021, is a word problem. I know how you feel about word problems. Please know that I support you in your feelings about word problems. 😊 Hear me out on this one... I find these kinds of word problems to be very straight forward. Therefore, I don't dislike them as much as other word problems. Once you know that you're dealing with a binomial distribution scenario, you're in good shape. Once you know the given info and non-given info, you're halfway there. Getting through the other half, isn't so bad. 

You will need, at least, a scientific calculator to assist you with the computations. To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to use the Binomial distribution formula
        ✔️     How to use your calculator
             

WMP! #68 says...

Happy solving!

P.S. There is nothing wrong with looking it up, even if you're not familiar with it.

Check back on Saturday, December 18th 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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To start this solution, the first thing I did was identify the pertinent information.

Here is a reminder of the formula for the Binomial Distribution:

Next, I take the pertinent information, substitute it appropriately in the formula, and evaluate. I did mention that you should, at least, have a scientific calculator; it's so that you could compute the combination.

I decided to include a computation using Desmos Scientific Calculator.

One last thing...the answer to the question: The probability that exactly 10 of 12 seeds planted will sprout is approximately 0.0702.

▪️ Have you ever worked with the Binomial Distribution before??

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

Thank you for solving with me this week!
Thank you for solving with me anytime this year! ❤️
I am a better problem solver now than I was at the top of 2021.

WMP! #69 is up next but won't be up until January 2022.
See you then and enjoy the holiday season.  😊

Cheers!

The Younge Lady

Weekly Math Problem #67

Expectation & Variance. I am definitely aware that if you're not in, or will not be undertaking certain disciplines in college, there will be no need for you to take upper-level statistics courses. Most students I encounter only need entry level statistics courses to satisfy math credits and/or as prerequisite for a non-math, discipline course. I was a math major in undergrad and am also a math major for grad...so I had the pleasure 🙃 of taking upper-level statistics courses. 

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

        ✔️     Properties of expectation
        ✔️     Properties of variance
     

Here is what WMP! #67 says...

Happy solving!

P.S. There is nothing wrong with looking it up, even if you're not familiar with it.

Check back on Saturday, December 11th 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 jump into this solution!

Hopefully, you've gotten a chance to look at the properties for expectation and variance involving random variables. If so, then you may have noticed that for part A, there were no formulas for expectation that involved variance. However, if you saw the properties for variance, then you should've seen a formula that involved expectation. That variance formula was taken and rearranged, so that expectation could be isolated.

I am aware that in isolating E(X), there should be a negative value also. I have chosen not to use it, since all of the givens are positive.

Parts B and C, are more straightforward when it comes to using the properties. Part B is dependent on the result from Part A.

▪️ Have you ever worked with expectation and variance in this way before??

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

Thanks for solving with me this week!
WMP! #68 is up next...the last WMP of 2021. 😥

Cheers!

The Younge Lady

Weekly Math Problem #66

Quadratic Inequality. I checked my blog and saw that I have not done a quadratic inequality yet...so, here we are. 🙂 In WMP #31, I solved an absolute value inequality. Make sure to check it out. 😉

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

        ✔️     Methods to solve a quadratic equation
        ✔️     Evaluating an expression
        ✔️     How to test an expression in an inequality

     

WMP! #66 wants us to...

Happy solving!

Check back on Saturday, December 4th 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 got started solving this week's problem by converting the inequality to an equation. I then solved the resulting quadratic equation by using the AC method for factoring. This process requires factoring by grouping.

Now that I have values for x, I continue to solve the original inequality by considering the three regions of the number line that the values create--the region to the left of 2/9, the region between 2/9 and 3, and the region to the right of 3. I then choose a value in each region to test in the original inequality. If the result of the tested value is greater than zero, that region is part of my solution. If the result of the tested value is less than zero, that region is not part of my solution.

According to the tested results, my solution is as follows:

...And that's a wrap! 

▪️ Were you able to solve this inequality??

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

Thanks for solving with me this week!
WMP! #67 is up next. 👩🏿‍🏫

Cheers!

The Younge Lady

Weekly Math Problem #65

Definition of a Derivative. One of the nice things about math is that rules just work. Without knowing why rules work, we are able to use them for our purposes. 😁 Derivatives are a classic example of this! I'm so glad that I can just use derivative rules to find the derivatives of functions, instead of using the definition of a derivative all the time. I remember learning precalculus/calculus and being introduced to and having to use the definition of the derivative. Once I learned derivative rules, I breathed a sigh of relief. The definition required lots of writing and, sometimes, fancy algebra tricks. 🙄

Despite my laziness, we're going to use the limit definition of the derivative this week. To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to use a formula (substitution)
        ✔️     How to find a limit
        ✔️     Simplifying fractions

     

WMP! #65 says...

Happy solving!

Check back on Saturday, November 27th 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 get into the solution for this week's problem! 

The solution requires us to use substitution with the limit definition of the derivative. Then, evaluate the limit.

Houston...we have a problem! When evaluating the limit, we end up with an undefined, rational expression. 🤔 This doesn't tell me anything about the derivative of the function. What is does tell me is that another technique is needed to evaluate the limit. 

What we're going to do is rationalize the numerator. How do I know that? ...From experience. Remembering this technique comes from repetition over the years.  Rationalizing the numerator will yield an equivalent expression that allows us to evaluate the limit.

Now, we can replace the equivalent expression and evaluate the limit.

Now we have a function for the first derivative of the square root of x. 👍🏿 

There is a quicker way to get to this result...and that's by using the power rule for derivatives. 

I don't know about you, but I am grateful for rules and shortcuts. 😁

▪️ Were you able to find the first derivative using the limit definition??

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

Thanks for sticking around and solving with me this week
...I truly appreciate it!
Of course, WMP! #66 is up next. 💪🏿

Cheers!

The Younge Lady

Weekly Math Problem #64

Cone Calculations. To lighten things up this week, I'm falling back on some Geometry. 😁 Sometimes, you just need to be reminded of how to do certain types of computation. I've definitely worked with a cone on many occasions (mathematically, not in real life), but can't really recall ever being asked to find the surface are of one. That's one of the things we'll be doing this week.

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

        ✔️     How to use a formula (substitution)
        ✔️     How to use your calculator
        ✔️     Rounding numbers

     

WMP! #64 wants us to...

Happy solving!

Check back on Saturday, November 20th 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: 

Just for fun, here is the online computation using the Desmos Scientific Calculator:

▪️ How did you feel about working with the surface area formula for a cone??

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

Thanks for solving with me this week!
Up next is WMP! #65 🤓

Cheers!

The Younge Lady

Weekly Math Problem #63

Proof by Induction. Proofs! 🙄 Learning them as an undergrad was so annoying! I just had the hardest time wrapping my brain around them, at the time. Proofs were also the reason I gravitated more toward applied problems. My mind had a much easier time understanding them. I knew proofs were important, but I decided other people will prove stuff and I'll be one of the ones to apply what was proven. 👍🏿 

Since I've started the blog posts, I haven't done any proofs. So, I thought this would be a good time to do one. I must admit that I had some recent practice when a student requested tutoring from me for Number Theory, where proof by induction is one of the topics. I met with the student and told the student to bring class notes to the session. I was able to see and understand the process better.

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

        ✔️     How to do a proof via induction method

     

WMP! #63 wants us to...

Happy solving!

Check back on Saturday, November 13th 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 are the steps I went through to complete the proof... 

Step 1: Define the function stated in the problem.

Step 2: Show that the function works for the first number in the set.

Step 3: Make an important assumption.

Step 4: Prove the function holds for n+1.

Step 5: Conclude the proof.

▪️ How did you feel about proofs??

▪️ Were you able to complete this proof??

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

Thanks for solving with me this week!
Up next is WMP! #64 🤓

Cheers!

The Younge Lady