solving the exponential equilateral triangle problem
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- เผยแพร่เมื่อ 20 ส.ค. 2024
- Solving the exponential equilateral triangle problem. Given an equilateral triangle with each side being e^x and its area being e^(3x), how can we solve for x? We need to use the rule of exponentials, the area of an equilateral triangle, and also the logarithm. Give it a try before you watch the solution! @bprpmathbasics
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We need you to moonwalk to the board one day!!!
Lol
Seconded
What about godbridging?
Using this formula will get you to the answer faster.
S=1/2×a×b×sin(c)
c is the middle angle of two sides a-b
We know that this is an equilateral triangle. So each angle will be equal to 60 degrees.
S=1/2×e^(2x)×sin(π/3)=e^(3x)
Simplify: e^(x)=sqrt(3)/4
X=ln(sqrt(3)/4)
Ecactly my thoughts
This method is valid, but it uses the fact that sin(π/3) = sin(60º) = (√3)/2. I believe part of the video's purpose is to show us how we can prove what the altitude (height) of an equilateral triangle is. Once we get to 2:16, I think it's just as fast to calculate using the triangle's area as it is to use the sine of 60º.
@@swsennel I know, his goal is to teach and it's good that he doesn't use too many formulas.
im gonna say . its the first time i solve one of your problems on my own without your help and im feeling awesome. i use sin60 degrees to find the height but apart from that i solved it and i feel amazing
Fun fact. We can make a formula for n face equilateral polygon for which side is e^x and area is e^(nx) where n is the number of faces (for example, a hexagon would have side e^x and area e^6x). The function is "f(n) = (ln(cot(π/n)) + ln(n) - 2ln(2)) / (n-2)" where n> 2. When we put the number of faces under n we get x of the given polygon, then we can put x under e^x to get the side length and under e^(nx) to get the area of the polygon. We can also find out from the derivative d/dn f(n) that the longest possible side has an equilateral octagon. Thanks for the idea
I got the answer using herons formula.
Great creativity. I like the problem better than the solution, the fact that you thought of an interesting mathematical idea.
It's also easy to do using Heron's formula.
Never liked the herons formula 😬
But in this case it simplifies good enough
Quickly recalling the formula for the height of an equilateral triangle, prove h=[sqrt(3) * s]/2. Plug in s=e^x and use the formula for area of triangle A=(b*h)/2, and since A=e^(3x) we have e^(3x)=[e^(2x)*sqrt(3)]/4 with simplifications. Divide by e^(2x) on both sides and end up with e^x=sqrt(3)/4 so x=ln(sqrt(3)/2).
Since x can be written as (ln3)/2-2ln2, I tried rewriting it like that too, but it just seemed more confusing. The value of x doesn't matter anyways because the side of the triangle (e^x) is always positive, so I'd say the simplest is best. Plus if you plug in x=ln(sqrt(3)/2) on e^x you get that s=sqrt(3)/2.
s=e^x??????????
@@quandaledinglegaming9930 s: side of tequila triangle. In this problem, s=e^x
@@nomic655 thanks
@@quandaledinglegaming9930 ???
Pretty neat lil problem.
Instead of doing the Pythagorean theorem stuff, you could have split this into two right angle triangles and applied 30 60 90 triangle knowing that it is an equilateral triangle we know that all sides are congruent which means that all angles are necessarily congruent.. That would save time but your way is active as well. Thank you sir
I think we use the Pythagorean Theorem to find out a formula for h and I don't know how can that happen with your method
@@nomic655 you know that if the base of the right triangle is (1/2)e^x and the hypotenuse is e^x then the height will be (sqrt(3)/2)e^x. This is because in a 30-60-90 the long leg is sqrt(3) times the short leg. I think that's what they were saying
@@rhymmasta8365 I'm pretty sure the way to prove that is with the Pythagorean Theorem
@@nomic655 is that the only way?
@@rhymmasta8365 Well, yes.
A= (e^x)*(sqr(3)/2) * (e^x)2 = (e^x)^3 … then divide by e^(2x) b/s
Sqr(3)/4 = e^x
x = 0.5*ln(3) - 2ln(2)
Very nice problem. It is also interesting that x is a neg value but e^x is positive
When the base for an exponent is positive the result is always positive even if the exponent is negative. This is because negative exponents are just the reciprocal of their positive counterparts.
i.e. x^(-y) = 1/(x^y)
or we couls just do the area of an equilateral wich is L^2 * square root 3 all divided by 4 and its pretty simple e^2x x square root 3 divided by 4 =e^3x , we divide by e^2x and then we take ln
Even though I like circles more than triangles (dunno why) but I'll consider this geo problem as a gift for my birthday today, so thank you :)
happy bday!
@@nomic655 Thank you :)
Happy birthday 🎉
@@equal7958 Thank you :)
I definitely like when answers are factored to primes.
Interesting problem, but the fact that your answer was a negative number for a geometric side length was… interesting, and I was kind of curious why you didn’t address that. Seems kind of important given what the problem context was, haha.
Thanks for the video, I did it myself the way you did it before looking at your solution, so I'm happy, but I found the result weird because sqrt(3)/4 is smaller than 1, so x=ln(sqrt(3)/4) will be a negative value. I was confused about having negative length but you did so, so why not 😵
@@andeew-4066 oh yeah true, I forgot about that, thanks
X can be a negative value and that doesn't matter. It's e^x to be the side length of this triangle and that is NOT a negative length.
one could rewrite the problem stating sides are y and area is y^3, then no exponentials and no confusion
@@deept3215 yep. This is junior middle school level.
This is a fun little problem.
h=e^xsin(60) =((3^1/2)/2)e^x using trigonometric
That square root symbol at 2:34 gave me nightmares
I’m sure you can generalise this for any base other than e and nonzero, non-one. For 2ⁿ it would be same result with log₂, for 3ⁿ it becomes log₃ for 5ⁿ log₅ etc.
Yep. It's the exponential function on the left to be equal to a constant sqrt(3)/4.
its nice how when you evulate ln(sqrt3/4) it equals -0.83...
05:10 Agreed, that makes it really cool. I'm serious -- and not in the way Biden says, "I'm serious," "It's no joke," etc. Your channel is top-notch, and your creativity is one reason why.
You americans just have to politicise everything, don't you? I can't imagine what it must take to shoehorn Biden and Trump into every conversation
@@marcushendriksen8415 It is often a form of mockery to "agree" with someone who said something self-congratulatory. Such mockery is done by someone who wants others to be as insecure as they are. Sometimes it is healthy to be self-congratulatory. I thought it was healthy of him to do so and I agreed with his assessment. I didn't bring Trump into this thread -- you did. I cited a familiar and obvious example of disingenuousness to distance myself from a perception that I was being disingenuous in my agreement and praise. It must be boring to be offended by analogies.
2:28
Спасибо за видео!!!
COS 30 = (sqrt3)/2
Strange how the area has units of Length^3 which is volume
I like your video very much. Just be careful whith your "4", they look like the Ψ (Psi) Greck or Cyrillic letter. It' a detail.
Awesome!!!
Nice one 👍👍
Thank you for everything
I need you to tell me the way how can I solve any problem in math
using your brain and hand
And practicing helps. A lot.
Don't give up on your first tries.
In other words we are looking for the side s of an equilateral triangle such as its are is equal to s^3. So:
s^3=s^2*sqrt(3)/4 => s=sqrt(3)/4. [=> x=ln(s)]
Done!
Nice
♥️♥️♥️♥️
Pls really need you to help me solve the question you gave last time
I can't do it
2^x + 3^x² =6
I would really appreciate
What are u saying
Show!!!
Can anybody tell me, when the pokemon had gone? Just noticed that the pokemon is missing...
Grt thnku
A=(1/2.5 x sqrt(3).5/2) / 2, why didn't you write / 2?
TIL that there's an equilateral triangle formula. I used Heron's formula. It got messy. I gave up.
I got it using the Heron's formula (for equilateral A = sqrt(3) a^2 / 4)
Heron formula has always been too long
Isn't ur final answer a negative number?
x can be a negative number, because the side length is e^x, which is positive, and the area is e^3x, which is also positive.
X is a negative number yes. But notice that X is in the "exponent"
A negative number in the exponent is no biggie coz all it does is reciprocate.
Is there any solution for |sin(x)|= |sin((180/π)*x)