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! #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! #25

Basic Probability. I believe this is the first probability problem I'm doing on the blog. I've had a love/hate relationship with probability in the past. Like a lot of other things in life, repetition helped over the years.Let's not talk about Probability Theory. 😣 When I was learning it, my brain took a long while to grasp it. When that course was over, I breathed a sigh of relief. Some things really do take time. In the the case of my brain and probability theory, it was taking more time than the semester would allow. 😅

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

        ✔️ How to use basic probability formulas

Yeah, that's it. Here is WMP! #25:

Happy solving!

Check back on Friday, October 30th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Well, once you get the correct formulas, you just need to make sure that you put the given values in the correct places in the formulas. (I can't tell you how many times I've seen that kind of mistake.) Here's how I worked out the questions:

Notice in part (iv), I gave an explanation why the events are independent, even though the question didn't ask, "why or why not?" It's not a coincidence that the answer from part (iv) is the same as the answer from part (i). When events are independent, one has no effect on the other. So, the probability of both events occurring is the same as multiplying their individual probabilities. Read more about independent events from MathisFun's page, Probability: Independent Events. 

 

▪️ Did you find these problems easy to solve??

▪️ How do you feel about probability??

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

Thanks for solving with me this week!

See you next week for WMP! #26

😁➡️

Cheers!

The Younge Lady

Weekly Math Problem! #19

Oblique Triangles. Back to some good ol' trig. 👍🏿 Different than special triangles, oblique triangles don't really have ways to come up with any of the values for the angles or legs of the triangles by following "shortcuts". You pretty much have to rely on the Law of Sines or Law of Cosines to get the answers to the problems. 

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

    ✔️  When/How to use the Law of Sines

    ✔️  When/How to use the Law of Cosines

    ✔️  How to find the area of an oblique triangle

    

By the way, you'll need a scientific calculator to help you in computing.

WMP! #19 says...

Happy solving! 

Check back on Friday, September 11th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Where to start? Well, there is not enough information to solve for any of the unknown angles, B or C. So, we'll solve for the missing side, a. One thing to notice is that the given triangle is not a right triangle, therefore, we cannot use the Pythagorean theorem. That's okay. We'll use something that looks a lot like it--the Law of Cosines formula. We'll use the one that fits our given information.

Now that all the sides are known, we have enough information to solve for the remaining angles, B and C. You have the option to use the Law of Cosines formula again, or you can use the Law of Sines. There's a catch...🤨. Sorry. 🤷🏿 . If you choose to use the Law of Sines, then you will run into a problem solving for angle B. Why?? "A triangle with two sides and a consecutive angle is the ambiguous case of the Law of Sines. In this case, you may have one, two, or no solutions, (Pre-Calculus Workbook for Dummies, Gilman, et al., 2009, p. 179)." Using the Law of Sines to solve for angle C is just fine. If you continue to solve for the remaining angles using the Law of Cosines, you also be fine. What did I do? I used the Law of Sines to solve for angle C.

Now that a second angle has been found, we can easily use subtraction to solve for the last angle:

It's done...the triangle has been solved for! Here's the triangle summary:

The last thing the needs to be done is to find the area of the triangle. Since the height of the triangle cannot be easily seen or found, we will not be using the traditional area formula of a triangle. Here's another formula to find the area of an oblique triangle:

The area can also be found by replacing the sides and angle, accordingly, in the formula. Similar results will be found.

 

◾️ How did you do in solving this week's problem?? 

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

Up next... WMP! #20❗️

Cheers!

The Younge Lady