Calling all students, teachers, and parents! As everyone is stuck at home during a global pandemic, now is a great time we are all forced to try and understand math (and our sanity level) a little bit more. Well, I may not be able to help you with the keeping sanity stuff, but as far as math goes, hopefully, the below math resources offer some much needed mathematic support.
All jokes aside I hope everyone is staying safe and successfully social distancing. Stay well, math friends! 🙂
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Study.com: In a time when companies are being more generous, Study.com is here for us as they offer up to 1000 licenses for school districts and free lessons for teachers, students, and parents. Check out all the education freebies here:
Math Planet: If you’re looking for free math resources in Pre-Algebra, Algebra, Algebra 2, and Geometry then you will find the answers you need at Math Planet. All free all the time, find their website here:
MathSux: Clearly, I had to mention MathSux, the very site you are on right now! Check out free math videos, lesson, practice questions, and more for understanding math any way that works for you!
Calling all NYC dwellers! Have you seen the new structure at Hudson Yards? A staircase to nowhere, this bee-hive like structure is for the true adventurists at heart; Clearly, I had to check it out!
Where does math come in here you say? Well, during my exploration, I had to wonder (as am sure most people do) what is the volume of this almost cone-like structure? It seemed like the best way to estimate the volume here, was to use the formula for the volume of a cone!
What do you think the Volume is?
Volume of a Cone:
I estimated the volume by using the formula of a three-dimensional cone. (Not an exact measurement of the Vessel, but close enough!) .
We can find the radius and height based on the given information above. Everything we need for our formula is right here!
Now that we have our information, let’s fill in our formula and calculate!
Extra Tip! Notice that we labeled the solution with feet cubed , which is the short-handed way to write “feet cubed.” Why feet cubed instead of feet squared? Or just plain old feet? When we use our formula we are multiplying three numbers all measured in feet:
radius X radius X (Height/3)
All three values are measured in feet! –> Feet cubed ()
Completing the Square: So many steps, such little time. It sounds like it involves a square or maybe this is a geometry problem? Why am I doing this again? Why must we complete the square in the first place?
These are all the thoughts that cross our minds when first learning how to complete the square. Well, I’m here to tell you there is a reason for all those steps and they aren’t that bad if you really break them down, let’s take a look!
Answer: x={-7,3}
Explanation:
I’m not going to lie to you here, there are a lot of seemingly meaningless steps to completing the square. The truth is though (as shocking as it may be), is that they are not meaningless, they do form a pattern, and that there is a reason! Before we dive into why let’s look at how to solve this step by step:
Feeling accomplished yet!? Confused? All normal feelings. There are many steps to this process so go back and review, practice, and pay close attention to where things get fuzzy.
But the big question is why are we doing these steps in the first place? Why does this work out, to begin with?
For those of you who are curious, continue to read below!
Want more Mathsux? Don’t forget to check out our Youtube channel and more below! And if you have any questions, please don’t hesitate to comment below. Happy Calculating!
Need more of an explanation? Check out why we complete the square in the first place here !
Ahoy math friends and welcome to Math Sux! In this post, we are going to learn about the equation of a circle, mainly how to write it when given a circle on a coordinate plane. We’ll see how to find the radius and center from the graphed circle and then see how to transfer that into our equation of a circle. And we’re going to do all of this step by step with the following Regents question. Stay curious and happy calculating! 🙂
How do I Answer this Question?
Let’s mark up this coordinate plane and take as much information away from it as possible. When looking at this coordinate plane, we can find the value of the circle’s center and the circle’s radius.
Now that found the center and radius, we can fit each into our equation:
Center=(-3,-4)
Radius=5
The only equation that matches our center and radius, is choice (2).
If the above answer makes sense to you, that’s great! If you need a little further explanation, keep going!
Equation of a Circle:
Equation of a Circle: Let’s take a look at our answer and break down what each part means:
So what do you think of circles now? Not to shabby, ehh? 🙂
Looking for more on circles? Check out this post on how to find the Area of a Sector here!
Howdy math peeps! In this post, we are going to go over the recursive formula step by step by reviewing an old Regent’s question. These things may look weird, confusing, and like a “what am I doing?” moment, but trust me they are not so bad! We are going to take a look at the Regents question below and then find the correct recursive formula that follows the given sequence by going through each answer choice. Before we begin to answer our questions though, we will first define and break down what a recursive sequence is. Also, be sure to check out the video at the end of this post for more examples and further explanation. Happy calculating!
Before we dive into the solution to this question, let’s first look at a recursive formula example and define what they are in the first place!
What is the Recursive Formula?
A Recursive Formula is a formula that forms a sequence based on the previous term value. All this means is that it uses a formula to form a sequence-based pattern. Recursive formulas can take the shape of different types of sequences, including arithmetic sequences (sequence based on adding/subtracting numbers) or geometric sequences (sequence based on multiplying/dividing numbers). Check out the example below:
a1=2 , an+1=an+4
How do You Solve a Recursive Equation?
When solving a recursive equation, we are always given the first term and a formula. We start by using the first given term of our sequence, usually represented as a1, and plug it into the given formula to find the value of the second term of the sequence. Then we take the value of the previous term (the second term in our sequence) and plug it into our formula again, to find the third value of our sequence…. and the pattern continues! The solutions we get from each step up forms a sequence. If this sounds confusing, don’t worry because we are going to look at an example! Take a look at the recursive formula below:
a1=2 , an+1=an+4
Now let’s take another look at our recursive formula, this time breaking down what each part means:
a1 always represents the first given term, which in this case is a1=2. Next, we plug in 2 for an into our formula an+4 to get (2)+4=6. This gives us our second term in the sequence, which is 6. Next, we plug in 6 into the formula to get (6)+4=10, which is the value of the third term in the sequence. And we continue the pattern, always taking the value of the preceding term! In this case, I just found the first few terms below, but the recursive formula can continue infinitely! Again , if this sounds confusing, please take look at the pattern below:
Terms of the Sequence: 2, 6, 10, 14, 18….
Notice all the terms of the sequence are circled in pink, forming a sequence of, 2, 6, 10, 14, 18!? This is what the recursive formula produces!
Now back to our Original Question:
Q: What recursively-defined function represents the sequence 3, 7, 15, 31,
(1) f(1) = 3, f(n + 1) = 28 (n) +3
(2) f(1) = 3, f (m + 1) = 28(0) – 1
(3) f(1) = 3, f(n + 1) = 28 (n) +1
(4) f(1) = 3, f (n + 1) = 38 (n) -2
How do I answer this question?
At first glance, all of these answer choices may look exactly the same as there are many recursive formulas to choose from. The first thing we need to do is to identify how each answer choice is different. Notice below, the section highlighted in green? This is what we will focus on for finding the correct recursive formula!
The question we are working with actually gives us a sequence and we need to find the recursive formula that works with it! Our goal with this question is to work backward to test out each recursive formula given to us until we get the correct sequence. To begin, let’s first identify each term in our given sequence.
As we go through each answer choice, we are looking for the recursive formula that gives us the above sequence 3, 7, 15, 31. Let’s start with choice (1) which happens to be a type of geometric sequence.
Right away we can see the sequence forming for choice (1) is 3, 11, … where the first term, 3, matches our original sequence, but the second term we get which is, 11, does not. This means we will need to move on and find the sequence of the next option, choice (2), which happens to be another type of geometric sequence. Let’s take a look:
For choice (2), we can see that the sequence we get is 3, 7, 127 which matches our given sequence for the first two terms 3, 7, but does not match the third term 127, when we need a 15 here (the original sequence is 3,7,15,31). Thus, we must move onward to the next recursive formula by testing out choice (3), which this time is an arithmetic sequence! Maybe we will have better luck!
This last option, choice (3) provided us with the same sequence we were originally provided within the original question, 3, 7, 15, 31. We have found our answer and now we can celebrate!
Recursive Formula Examples
Still have questions about recursive formulas? Check out more on recursive formula examples here and in the video above! And if you’re looking to learn all there is to know about sequences, check out this post here! Looking to move ahead? Check out the infinite geometric series lesson here! Also, please don’t hesitate to comment with any questions. Happy calculating! 🙂
Ahoy math friends! In this post, we are going to focus on solving log equations by solving this Regents questions step by step. We’ll answer this question right away! But if you need more of a review, keep reading and you will find what logarithms are, the different kinds of logarithms rules, and some simpler examples. Ready for our first example?! Check it out below:
How do I Answer this Question?
Step 1: Let’s re-write the equation to get rid of the “log.”
Step 2: Solve for x in our new equation (5x-1)(1/3)=4
If the above answer makes sense to you, great! If not, that’s ok too, keep reading for a review on solving log equations.
What are Logarithms?
Logarithms are inverses of exponential equations. Take a look below for a clearer picture.
Basic Formula:
Logarithms can be re-written to get rid of the word “log.” This makes them easier to solve and understand.
Log Rules:
There are a few rules you have to memorize get used to with practice. These rules are used when solving for x in different kinds of algebraic log problems:
Still got questions? No problem! Check out the videos below and the post here for more on logarithms! Also don’t forget to subscribe below to get the latest FREE math videos, lessons, and practice questions from MathSux. Thanks for stopping by and happy calculating! 🙂
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