Weekly Math Problem! #44

Exponential and Logarithmic Equations. I was just fishing through some of my math folders to see what I had available that I could work on. You would think with all of the options I have that it would be easy to choose a problem. Actually, it was harder to make a choice; with so many options from various areas of math, it took me quite a while to settle on something. (You should see the amounts of files I have related to mathematics. 😄) So, I settled on the one below. 🤷🏿‍♀️  

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

        ✔️     How to solve an exponential equation
        ✔️     Logarithmic rules
        ✔️     How to solve a logarithmic equation

No calculator necessary. Without further ado, here is what WMP! #44 wants us to do:

Happy solving!

Check back on Friday, April 2nd 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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Hey y'all...can you believe it's Friday already? Yeah...me either. Without further ado...

Part (i)

Part (ii)

 

▪️ Did you get through these problems?

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

Thanks for solving with me this week!

Up next WMP! #45. 👩🏿‍🏫

Cheers!

The Younge Lady

Weekly Math Problem! #43

Universal Gravitation. Do you remember the molality problem from WMP! #40? (Check it out, if you haven't.) Well...I liked working on it and told myself that I need to incorporate more science-based problems into the weekly rotation. So, I'm doing another science problem this week. I want to begin re-familiarizing myself with the math involved in the sciences. So, I am excited to tackle physics this week. 😊 The last time I took physics was in high school, actually. So.....it's been way too long since I've done anything physics related--decades. This should be interestingly fun. 🥴  

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

        ✔️     How to rearrange a formula to isolate the variable of interest
        ✔️     How to work with scientific notation
        ✔️     How to work with units

You should definitely have a calculator handy, by the way. Here is WMP! #43:

Happy solving!

Check back on Friday, March 26th 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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So....hmmm.... let's see 👀 what's going on here. When you look up "Universal Gravitation", search results will include the Law of Universal Gravitation, images with circles (that represent two objects be it plants, people, or other), and the formula for it. For the purposes of focusing on the mathematical component, we'll just be focusing on the formula. Here is the formula and what each variable means with their corresponding units:

Now that we know the parts of the equation, let's examine the problem to see which information is provided for us to solve the problem. In true Younge Lady fashion, I've color-coded the information.

As you can see, according to the way I've labeled the man and the moon, we need to solve for the mass of the moon. The way the formula is given won't work for us, so we need to rearrange it to isolate the variable m2. What's nice about the formula is that it is a proportion. So, all I need to do is multiply both sides by the right expression that will allow me to isolate m2. Here goes...

Now, we can make the appropriate substitutions and solve for the mass of the moon. If you have a really cool scientific or graphing calculator with a good display, you can plug all of this in and compute the answer in one step. 

**This computation was done with Desmos Scientific Calculator. Click image to enlarge.

The mass of the moon is 7.359x1022 kg. I hope you found this helpful. It was nice working on the problem this week.

 

▪️ Did you have any trouble solving for the mass of the moon?

▪️ When was the last time you took a physics class (if you ever did)?

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

Thanks for solving with me this week!

Now...onto WMP! #44. 👊🏿

Cheers!

The Younge Lady

Weekly Math Problem! #42

Sampling Distribution. This week's problem is inspired a student I'll be working with this week. Statistics presents itself in various subjects outside of mathematics. A common topic of students learning statistics is computing and using a z-score. A z-score helps to standardize data so that probabilities related to that data can be found...at least that's how my brain 🧠 thinks about it. Age, salaries, test scores, etc. If it can quantified and certain other information (mean and/or standard deviation) about the data is known, then you may be able to find and use z-score. ...And you know what that means...word problems! 😩

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

        ✔️     How to calculate the z-score
        ✔️     How to use a Normal probability table

Here is WMP! #42:

Happy solving!

Check back on Friday, March 19th 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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Alright, alright. Because word problems can be annoying, I always advise to identify the information in the problem that you will be using to solve it. So, here is my color-coded version of the problem, highlighting the important information:

Next, we'll use the highlighted information, above, in the z-score formula. The only way we'll be able to find the probability of interest is to standardize the values of interest. Since there are two values of interest, we compute two z-scores. This will allow us to use a Normal probability table.  

 

Now that we have the two z-scores we need, we can find probability of interest. You can definitely use a wide variety of software and probability calculators to obtain the answer. I used a couple and will share those screen shots soon. First, let's use a Normal probability table. There are many to choose from. The table I linked has a simple display, but you still need to understand the symmetrical aspect of a Normal curve to use it.

The problem asked for "the probability that the average blood pressure...will be between 118 and 124 mmHg." Because we standardized the data, we can look at this as the probability between the z-scores -0.4 and 0.2

One of our z-scores is negative and the other is positive. The Normal table being used here only shows positive z-scores. Here is where the beauty of the symmetrical nature of the Normal curve is helpful. I will "switch" the negative z-score with its positive counterpart so that I can use the probabilities from the table.

As promised, here are a few screen shots of online probability calculators I used that confirmed the answer above:

**This plot was generated using Geogebra's Classic Probability Calculator. Click image to enlarge.

**This plot was generated using Online Stat Book's Area Under Normal Distribution Calculator. Click image to enlarge.

**This plot was generated using MathCracker's Normal Probability Grapher. Click image to enlarge.

So to answer the question... The probability that the average blood pressure of this sample will between 118 and 124 mmHg is 0.2347. 👍🏿

 

▪️ How did you go about finding the probability?

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

Thanks for solving with me this week!

Up next WMP! #43. 👩🏿‍🏫

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