Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

mathmate

QUIZ TIME : easy question : area of triangle The plane 6x+4y+2z=12 intersects the coordinate planes to form three sides of a triangle. Find the area of the triangle. The challenge here is to show as many ways as you can find to get the correct answer. Give exact values if you can, eg. sqrt(14) instead of 3.742. NOTE to readers: Please award a medal to someone who presents a formula or method that you would not have used.

  • 2 years ago
  • 2 years ago

  • This Question is Closed
  1. mathmate
    Best Response
    You've already chosen the best response.
    Medals 1

    • 2 years ago
    1 Attachment
  2. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    x=0, get the intercepts; put them in a distance formula .... di this for each variable to zero out. then you have the side measures and can do a heron on it

    • 2 years ago
  3. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    or... divide by 12 and divide off the tops to get the intercepts if we dont have time to do them one by one

    • 2 years ago
  4. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    6x+4y+2z=12 12 12 12 /6 /4 /2 ------------- 2 3 6 are our intercepts

    • 2 years ago
  5. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    one way is we can take these and form vectors to get an angle to do a sin area formula with

    • 2 years ago
  6. arijit.mech
    Best Response
    You've already chosen the best response.
    Medals 0

    3.164 squre unit approx using area = squrt{ s(s-a)(s-b)(s-c)) ;s=(a+b+c)/2

    • 2 years ago
  7. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    (2,0,0) (0,0,6) -(0,3,0) -(0,3,0) ------- ------- <2,-3,0> <0,-3,6> cos(yaxis) = <2,-3,0>.<0,-3,6> 9 ----------------- = --------------- |<2,-3,0>| |<0,-3,6>| sqrt(4+9+9+36) cos(y) = 9/sqrt(58) y = cos-1(9/sqrt(58)) |<2,-3,0>| |<0,-3,6>| sin(cos-1(9/sqrt(58))) / 2 9/sqrt(58) sin(cos-1(9/sqrt(58))) / 2 maybe :)

    • 2 years ago
  8. mathmate
    Best Response
    You've already chosen the best response.
    Medals 1

    So far I see Heron's formula from Amistre64 and arijit, and vectors from Amistre64. Please provide detailed calculations and answers.

    • 2 years ago
  9. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    my error is in the sqrt(58) :) sqrt(4+9) * sqrt(9+36) sqrt(13*45) 3 sqrt(65)

    • 2 years ago
  10. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    Base * (..... ......height..........) /2 9/3sqrt(65) sin(cos-1(9/3sqrt(65))) / 2 3/sqrt(65) sin(cos-1(3/sqrt(65))) / 2 hopefully lol

    • 2 years ago
  11. arijit.mech
    Best Response
    You've already chosen the best response.
    Medals 0

    PLEASE SEE THE DRAWING........

    • 2 years ago
    1 Attachment
  12. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    i got a few more errors, but the concepts seems solid ;)

    • 2 years ago
  13. mathmate
    Best Response
    You've already chosen the best response.
    Medals 1

    So 2,3,6 are the intercepts, sqrt(13), sqrt(45), sqrt(40) are the sides.

    • 2 years ago
  14. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    right, and you can heron that for sheer simplicity

    • 2 years ago
  15. mathmate
    Best Response
    You've already chosen the best response.
    Medals 1

    Some "exact" answers would be nice! :)

    • 2 years ago
  16. mathmate
    Best Response
    You've already chosen the best response.
    Medals 1

    Agree, we also want as many different ways as possible.

    • 2 years ago
  17. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1324567464017:dw|

    • 2 years ago
  18. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    sqrt(45) sin(y) = height sqrt(13) = base

    • 2 years ago
  19. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    b*h/2 = 3sqrt(14) if i did it right that time

    • 2 years ago
  20. mathmate
    Best Response
    You've already chosen the best response.
    Medals 1

    Wow! First exact answer! 3sqrt(14) (that's what I have too). So far we have intercepts: 2,3,6 sides : sqrt(13), sqrt(45), sqrt(40) Area : 3sqrt(14) Methods: Heron : Amistre64, arigit Vectors: Amistre64 (to be completed) bh/2 : Amistre64

    • 2 years ago
  21. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    the bh/2 is after the vectors give us a sin for an angle to determine the height

    • 2 years ago
  22. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    i wonder if its worth it to do a 3d integration :) or at least take the measures and translate them to a xy congruent triangle

    • 2 years ago
  23. phi
    Best Response
    You've already chosen the best response.
    Medals 2

    \[A= \frac{1}{2}\sqrt{|x|^2|y|^2-(x \cdot y)^2}\] with (using amistre's two vectors) x=<2,-3,0> y= <0,-3,6> gives 0.5sqrt( 13*45-81)= 1.5sqrt(65-9)= 1.5sqrt(56)= 3sqrt(14)

    • 2 years ago
  24. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    work in progress here |dw:1324568575430:dw| \[\int_{0}^{6}distance.x.to.y\ dz\]

    • 2 years ago
  25. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    \[X_i=\left(\frac{z-6}{3},0,z\right)\] \[Y_i=\left(0,\frac{z-6}{2},z\right)\] \[X_i-Y_i=\left(\frac{z-6}{3},\frac{-z+6}{2},0\right)\] distance from X to Y:\[\sqrt{(\frac{z-6}{3})^2+(\frac{-z+6}{2})^2}\] \[\sqrt{\frac{z^2-6z+36}{9}+\frac{z^2-6z+36}{4}}\] \[\sqrt{\frac{4z^2-4.6z+4.36+9z^2-9.6z+9.36}{36}}\] \[\frac{\sqrt{13z^2-78z+468}}{6}\] ergo lol \[\int_{0}^{6} \frac{\sqrt{13z^2-78z+468}}{6}dz\]

    • 2 years ago
  26. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    hmm, wolf says that about 19, which is a bit off from the 3sqrt(14) i wonder why

    • 2 years ago
  27. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    348 not 468 ... but still thats off

    • 2 years ago
  28. amistre64
    Best Response
    You've already chosen the best response.
    Medals 1

    oh well, it was a thought. guess i dont know how to freehand integrals yet :)

    • 2 years ago
  29. mathmate
    Best Response
    You've already chosen the best response.
    Medals 1

    OK, here's what we have so far (let me know if I missed anything) Phi: does the formula come from cross product, it's neat, like a generalized form of bh/2. Related to that, I suggest the cross product of Amistre64's two vectors. (1/2)AxB=(1/2)<0,-3,6>x<2,-3,0>=(1/2)sqrt(504)=3sqrt(14) Methods: 1. Heron : Amistre64, arigit 2. (1/2)xysin(theta) + bh/2 : Amistre64 3. integration : Amistre64 (in progress) 4. (1/2)sqrt(|x|^2|y|^2-x.y) : phi 5. half magnitude of cross-product, (1/2)|PxQ|: mathmate

    • 2 years ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.