Weekly Math Problem! #22

Differentiation Application. It's about time for another word problem. Yeah...I said it. 🤷🏿 Since I love me some calculus, the problem is coming from there. Solving word problems has never been a super strength, but I get better at them the more I do them. So, here we are...I need to get better.

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

        ✔️ How to translate English into math 🧐

        ✔️ How to find derivatives

Here is what WMP! #22 says:

Happy solving!

Check back on Friday, October 2nd for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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This is a related rates word problem. Related rates or not, I read word problems at least twice...most often three to four times. After reading the problem, the formula with which you will take its first derivative with respect to time needs to be identified. In this case, it will be the volume of a cone. Check the problem for any special relationships and adjust the formula accordingly. Then, take the first derivative with respect to time.

Now, we need to identify in the problem the givens and the quantity of interest. Here is the problem, once again, with the important information color-coded:

Let's plug stuff in! 🤣 Plug-in the given information from the problem and finish solving for quantity of interest. Don't forget about the units!

Well, according to the solving, the height of the pile is approximately increasing at a rate of 0.196 ft/min.That's what I got. How about you?  

 

▪️ How do you feel about related rates problems?? 

▪️ 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! #23

👍🏿

Cheers!

The Younge Lady

Weekly Math Problem! #21

Integration. So, here we are again with another integration problem. If you've done last week's problem, then you've already done some of the work needed to do this week's problem. 🤓 Wait, you haven't seen last week's problem?! Pause....check out WMP! #20. 

Last week I mentioned that partial fraction decomposition is an important technique that allows you to "simplify" some complicated rational expressions by breaking them down into smaller parts. WMP! #21 is such a problem that requires the technique of partial fraction decomposition.

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

        ✔️ How to integrate rational expressions

        ✔️ Partial fraction decomposition

WMP! #21 wants us to...

Happy solving!

Check back on Friday, September 25th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Again, if you take a look at WMP! #20, you will see that it has the same rational expression as this week's problem. So, when you're given a rational expression, similar to that one, to integrate, you can see what needs to be done to carry out the integration. Here's how I carried it out:

That's all I've got! (I actually really like this problem.)  

 

▪️ How did you feel about this week's problem?? 

▪️ Did you carry out the integration differently than I did??

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

See you right back here next week for WMP! #22, the last problem of September.

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Cheers!

The Younge Lady

Weekly Math Problem! #20

Partial Fractions Decomposition. This topic is one I didn't hear much about until I was required to integrate certain types of rational expressions. It's an important technique that allows you to "simplify" some complicated rational expressions by breaking them down into smaller parts, so that you may more easily compute problems that involve them. This week's problem comes from the Pure Mathematics, CAPE (Caribbean Advanced Proficiency Examination) materials.

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

        ✔️  How to use partial fraction decomposition

        ✔️  Simplifying rational equations

        ✔️  Working with undetermined coefficients

    

WMP! #20 says...

Happy solving! 

Check back on Friday, September 18th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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...And we're back to decompose this fraction into the components that were combined to get it in the first place. Here are the steps I took to do so:

There you have it!  

 

▪️ Were you able to decompose the fraction?? 

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

See you right back here next week for WMP! #21

(Now let's see if you can guess what next week's problem is going to be.)

🤓

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