Math Help

Discussion in 'Odds & Ends' started by bamboo711, Aug 27, 2008.

  1. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    Guys, I can't figure this question out: A spherical balloon with radius r inches has volume defined by the function below. Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 5 inches. (Give the answer in terms of π and r.)

    oh and the originial formula is v= 4/3 pi r^2
     
    #1
  2. ScovilleJenkins

    ScovilleJenkins Rookie

    Joined:
    Jul 1, 2005
    Messages:
    261
    old v=4/3pi r^2
    new v=4/3pi (r+5)^2 = 4/3 pi(r^2 +10r+25) = 4/3pir^2 + 40/3pi*r +100/3 pi

    air required = new v -old v = 40/3 pi*r + 100/3 pi = (20/3)pi (2r+5)

    its been a while since ive done something like this but i think its right
     
    #2
  3. Douggo

    Douggo Semi-Pro

    Joined:
    Feb 22, 2005
    Messages:
    549
    Location:
    Minnesota
    You're looking for the difference in volumes between radius (r+5) and radius r. Just plug them into the original formula and subtract. Not sure what "n" is supposed to be...
     
    #3
  4. gj011

    gj011 Banned

    Joined:
    Aug 21, 2007
    Messages:
    6,548
    Location:
    Back from prison
    Formula for sphere volume is:

    V = 4/3pi r^3.
     
    #4
  5. Steady Eddy

    Steady Eddy Hall of Fame

    Joined:
    Jul 19, 2007
    Messages:
    2,545
    Location:
    Arizona
    The answer is the difference between the sphere with an 'r + 5' radius, and the sphere with the 'r' radius.
    v = 4/3 pi (r + 5)^3 - 4/3 pi r^3
    v = 4/3 pi [(r + 5)^3 - r^3] factor out 4/3 pi
    v = 4/3 pi (r^3 + 15r^2 + 75r + 125 -r^3) expand (r + 5)^3
    v = 4/3 pi (15 r^2 + 75 r + 125) r^3 and minus r^3 cancel
     
    #5
  6. xtrakewl

    xtrakewl New User

    Joined:
    Aug 4, 2008
    Messages:
    93
    Location:
    canada
    the answer is 22
     
    #6
  7. Steady Eddy

    Steady Eddy Hall of Fame

    Joined:
    Jul 19, 2007
    Messages:
    2,545
    Location:
    Arizona
    22? Since the radius was stated in inches, and we're talking about volume, let's suppose that you mean 22 cubic inches. Sometimes the answer is a constant, like 22, and no variable r appears in the final answer. Rather than repeat my calculations above, let's see if '22' works if the original radius was 10 inches. After all, if the answer is 22, that means 22 works for ALL values of r.
    The original volume would be 4/3 pi 10^3. Which equals 4,188.79 cubic inches, (rounding to the nearest hundredth). The increased sphere would have a volume equal to 4/3 pi 15^3. Which equals 14,137.17 cubic inches. The difference is nearly 9,000 cubic inches, so the suggestion of 22 cannot be correct.
     
    #7
  8. xtremerunnerars

    xtremerunnerars Hall of Fame

    Joined:
    Sep 29, 2005
    Messages:
    1,640
    Location:
    Right behind you, but i'll be gone before you fini
    I think he was just being funny.
     
    #8
  9. YULitle

    YULitle Hall of Fame

    Joined:
    May 23, 2005
    Messages:
    4,133
    Location:
    Guymon, OK
    Yeah, isn't that supposed to be the answer to The Question in Hitchhiker's Guide?
     
    #9
  10. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    Thanks for the help guys. I eventually got it because of you. Thanks.

    I have another question:
    Find each of the following functions.
    f(x) = sqrt(2 - x)
    g(x) = sqrt(x^2 - 1)

    f+g
    f-g
    fg
    f/g

    I have no clue where to start. Help appreciated. Thanks!
     
    #10
  11. Steady Eddy

    Steady Eddy Hall of Fame

    Joined:
    Jul 19, 2007
    Messages:
    2,545
    Location:
    Arizona
    Who are you thanking? Me? Anyway, this is mostly a question to see if you know what the terminology means. For example, f + g = f(x) + g(x). See? So just replace f(x) with sqrt(2-x) and g(x) with sqrt(x^2 - 1). (You won't be able to combine like terms because of the "sqrt"s).

    f - g, fg, and f/g, work the same way. It's just about knowing the terminology, because these don't simplify into something that looks different.
     
    #11
  12. Sean Dugan

    Sean Dugan Rookie

    Joined:
    Feb 18, 2004
    Messages:
    163
    Take a pin and puncture the balloon. Ergo, Volume = 0.
     
    #12
  13. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    Steady Eddy, you are a math whiz pal. You got both of those right so thanks a lot for the help everyone, but especially Eddy.

    Any clue about this one?

    If you invest x dollars at 7% interest compounded annually, then the amount A(x) of the investment after one year is A(x) = 1.07x. Find each of the following.

    A of A

    A of A of A

    A of A of A of A

    Find a formula for the composition of n copies of A.


    Thanks guys!
     
    #13
  14. Lakoste

    Lakoste Professional

    Joined:
    May 15, 2005
    Messages:
    844
    That is 42
     
    #14
  15. Swissv2

    Swissv2 Hall of Fame

    Joined:
    Oct 19, 2005
    Messages:
    2,491
    Location:
    Tennis Courts!
    While I am sure they don't mind helping you out, I would recommend that you do not continually ask for direct answers to your homework, or those who wish to respond to the question can give an example problem related to the question.



    We will be happy to help out, but not do your homework for you.
     
    #15
  16. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    Honestly I got an answer but it's not correct. I can't come up with hte formula to calculate the percentages for each additional year. Any help?
     
    #16
  17. Bundey

    Bundey Professional

    Joined:
    Apr 28, 2007
    Messages:
    1,123
    Go play tennis
     
    #17
  18. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    School comes before tennis, sadly
     
    #18
  19. Bundey

    Bundey Professional

    Joined:
    Apr 28, 2007
    Messages:
    1,123
    What math is this?
     
    #19
  20. sapient007

    sapient007 Semi-Pro

    Joined:
    Jun 28, 2006
    Messages:
    582
    use this logic:

    A = annual balance
    x = interest

    starting year = A
    year 1 A(1 + x)
    year 2 A(1 + x)2
    etc
    etc
     
    #20
  21. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    Wait so what? I multiple the year x annual balance (1 + interest) ????

    An example maybe? Thanks!
     
    #21
  22. bumfluff

    bumfluff Semi-Pro

    Joined:
    May 1, 2007
    Messages:
    521
    Location:
    South England/Scotland
    You familiar with exponential growth?
     
    #22
  23. bumfluff

    bumfluff Semi-Pro

    Joined:
    May 1, 2007
    Messages:
    521
    Location:
    South England/Scotland
    Looking at the way you described the problem, I don't think that will help you with your understanding of what is being asked.

    You say x is the initial investment and yet you have the function A(x), however x is a constant so A(x) isn't really a function.

    A better way to look at it is to call y the amount you have in your account (I use y simply because if I were to graph the function then the amount would be on the y-axis so the use of x would be confusing and far too non-conformist for me) and yo the initial investment. We shall then say the amount in your account is equal to a function of the number of years (n) it has been in your account. so y = A(n). What you are essentially being asked is to find A(n) when the interest is 7%. I think you need to understand this first as it seems that you are not entirely sure what you are needing to do.
     
    #23
  24. Steady Eddy

    Steady Eddy Hall of Fame

    Joined:
    Jul 19, 2007
    Messages:
    2,545
    Location:
    Arizona
    Since A(x) = 1.07x
    A of A = 1.07(1.07x) = 1.07^2 x
    A of A of A = 1.07(1.07^2 x) = 1.07^3 x

    so A1 of A2 of ... of An = 1.07^n x

    so by using the exponents button on a scientific calculator, you can model compound interest.
     
    #24
  25. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    Thank you sir. I am having some difficulty with inverse functions:

    If g(x) = 6 + x + e^x, find g-1(7).

    AND

    Find a formula for the inverse of the function.
    f(x) = e^x^7

    Any ideas? Thanks guys (Eddy)
     
    #25
  26. meowmix

    meowmix Hall of Fame

    Joined:
    Jul 5, 2007
    Messages:
    3,852
    Location:
    Hanover, NH
    First, what math are you in? You shouldn't be having a problem with this...

    I won't give you the answer, I'll leave that for you to figure out. But for inverse functions, take the f(x)/g(x)/whatever(x) and put a y in place of it. Now for every x in the original equation, put a y and for every y, put an x. Solve for y. That's the inverse function. After that, it's just plug and chug.
     
    #26
  27. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    I'm in Calc.

    I know how to do simple problems but I mean significantly harder with various e^x and such.

    Like for y = e^x^7 I do ln to each side and get ln(y)= x^7.... then what?

    Thanks!
     
    #27
  28. topspin2

    topspin2 New User

    Joined:
    Jun 12, 2008
    Messages:
    89
    Location:
    On the court
    that be 42.
     
    #28
  29. Steady Eddy

    Steady Eddy Hall of Fame

    Joined:
    Jul 19, 2007
    Messages:
    2,545
    Location:
    Arizona
    let g-1(7)=x, by definition, g(x)=7, replace g(x) with 7, and re-write.
    7=6 + x + e^x
    1=x + e^x
    from here it seems obvious that x is zero. Maybe there is a deductive path, or maybe they expect you to be able to tell that it is zero just by looking at it.
    Finally, let's check, g(0) = 6 + 0 + e^0 = 7. It checks!

    let y = (e^x)^7, now we do the 'switcheroo'
    x = (e^y)^7 and our job now is to solve for y
    ln x = 7 ln(e^y)
    ln x = 7 (y)
    y = (1/7)(ln x)
    so f-1(x) = (1/7)(ln x)
     
    #29
  30. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    Eddy it was the 7 root of ln (x) but I ended up figuring that part out. So thanks.

    Any idea with this?

    Consider the following limit.
    lim_(x->-4) (4 x^2 + a x + a + 44)/(x^2 + 3 x - 4)
    (a) Find the number a such that the limit exists.
    (b) Find the value of the limit.


    Thanks guys.
     
    #30
  31. Steady Eddy

    Steady Eddy Hall of Fame

    Joined:
    Jul 19, 2007
    Messages:
    2,545
    Location:
    Arizona
    I wasn't wrong. They're the same thing. With a log you can make the exponent the coefficient and vice-versa. (1/7) as an exponent means to take the 7th root.

    First check to see that the denominator is zero when you plug in a -4 for x. Yep, it is. Ok, can we factor the denominator? (x + 4)(x - 1). This means we want to factor the numerator in such a way that "x + 4" is a factor of it as well. Let's use synthetic division.
    -4] 4 a (a + 44)
    -16 -4a+64
    4 a-16 -3a+108
    So we end up with a remainder of "-3a + 108", for a to be a factor, this remainder should be zero. So -3a + 108 = 0 which implies a = 36.

    So the answer to (a) is 36.
    Now we can write this as: 4x^2 + 36x + 80
    4(x^2 + 9x + 20), so 4(x +4)(x + 5)
    Since (x + 4) is in both the numerator and denominator, they cancel.
    [4(x + 5)] / [(x - 1)], since x approaches -4, we get
    4/(-5) or -4/5. So thats the answer to (b)
     
    #31
  32. Steady Eddy

    Steady Eddy Hall of Fame

    Joined:
    Jul 19, 2007
    Messages:
    2,545
    Location:
    Arizona
    Is that the answer you wanted? I didn't figure that out for nothing, did I?
     
    #32
  33. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    Yeah dude that was exactly right. How are you so good at all this? haha math major or something.

    Any clue with this?

    For the limit below, find values of δ that correspond to the ε values.
    lim_(x->0) (e^x - 1)/x = 1

    0.5 _____
    0.1 ______

    AND


    For the limit below, find values of δ that correspond to the M values.
    lim_(x->pi/2) tan^2(x) = infinity
    M
    500
    10,000




    Thanks steady eddy!
     
    Last edited: Sep 15, 2008
    #33
  34. Steady Eddy

    Steady Eddy Hall of Fame

    Joined:
    Jul 19, 2007
    Messages:
    2,545
    Location:
    Arizona
    You are welcome sir. This is kind of fun for me. You guessed it, math major.

    | (e^x - 1)/x - 1 | < .5
    -.5 < (e^x - 1)/x -1 < .5
    .5 < (e^x - 1)/x < 1.5 adding one to each part
    .5x < (e^x -1) < 1.5x multiplying each part by x
    .5x + 1 < e^x < 1.5x + 1 adding one to each part, again
    solve .5x + 1 < e^x and also solve e^x < 1.5x + 1
    x>-1.594 x<.762
    so -1.594 < x - δ < .7627
    so | x - δ | < .7627 is sufficient, since x => 0, we can say δ = .7627

    tan^2 (x) = 500
    tan(x)= radical(500) taking the square root of each side
    arctan(radical 500) = x = 1.526

    tan^2(x) = 10,000
    tan(x)= 100
    arctan(100)=x=1.5607966
    BTW, the actual value of pi/2 is 1.570796, pretty close, and you see how it gets close to the actual value as M gets larger.
     
    #34
  35. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    Steady Eddy,
    Thanks man you really helped me out with the first problem. Yeah you're pretty good at math to be able to do all these. Now for the second problem, you have done something wrong somewhere. For the part that is 500 I know the answer is .0447 somehow. Any ideas? Oh and I can't figure out the IV and V below. I think the first one is +-1 but I am not sure. Any ideas? The rest of problem below is where you're supposed to obtain the info from.

    A crystal growth furnace is used in research to determine how best to manufacture crystals used in electric components for the space shuttle. For proper growth of the crystal, the temperature must be controlled accurately by adjusting the input power. Suppose the relationship is given by the following equation, where T is the temperature in degrees Celsius and w is the power input in watts.
    T(w) = 0.1w2 + 2.156w + 20

    (a) How much power is needed to maintain the temperature at 201°C? (Give your answer correct to 2 decimal places.)


    (b) If the temperature is allowed to vary from 201°C by up to ±1°C, what range of wattage is allowed for the input power? (Give your answer correct to 2 decimal places.)

    (iv) What value of is given?
    _______°C

    (v) What is the corresponding value of ?
    ______watts



    Thanks for the help.
     
    #35
  36. Steady Eddy

    Steady Eddy Hall of Fame

    Joined:
    Jul 19, 2007
    Messages:
    2,545
    Location:
    Arizona
    I only have time right now to answer this first part. I'll get to the 'furnace' problem later. Here's how they got the 0.0447, recall that the answer I got for M = 500 is 1.526. How close it that to pi/2? That's 1.5707 - 1.526 = 0.0447. For the second part you'd get 1.5707 - 1.5607 = 0.01. Sorry for not following through. Later, dude.
     
    #36
  37. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    Thanks man... that was right. Yeah I'm not exactly certain about the last one. I'm pretty sure the IV part is +/-1 but I'm not sure at all what to put in for v.
     
    #37
  38. diredesire

    diredesire Super Moderator

    Joined:
    Mar 16, 2004
    Messages:
    6,720
    "somehow know the answer" == you have the solutions already? :twisted:
     
    #38
  39. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    I wish I had the solutions man. I did that problem like a week ago and just couldn't remember how to do it. haha. Steady Eddy is a beast at math though. Oh calculus......
     
    #39
  40. diredesire

    diredesire Super Moderator

    Joined:
    Mar 16, 2004
    Messages:
    6,720
    Is this a high school or college course? What book are you using, out of curiosity? :shock:
     
    #40
  41. Steady Eddy

    Steady Eddy Hall of Fame

    Joined:
    Jul 19, 2007
    Messages:
    2,545
    Location:
    Arizona
    To solve (a) means that .1w^2 + 2.156w + 20 = 201
    so, .1w^2 + 2.156w -181 = 0 (let's multiply by 10)
    w^2 + 21.56w - 1810 = 0
    the positive root is, w = 33.11 (rounding to two decimals)

    For (b) proceed like above, but solve for 200, and 202 (+/- 1)
    You will get 32.99 and 33.22 for your positive roots, respectively.

    I'm not too sure what they mean in IV) and v), but I have some ideas. For IV) they might only mean what is +/- 1 for 201, answer 200 and 202.

    For V), what are the w's associated with 200 and 202? answer 32.99 and 33.22, respectively.
    Keep pacticing that math. Lot's of practice, that's the 'secret'. Just like in tennis!
     
    #41
  42. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    This is college calculus. My book is Stewart Multivariable maybe? I read that it's not a very good book.

    Thanks Eddy, you helped me out greatly with that last question. I'm trying to get ahead of the game a little now, so how would you do a problem like this:

    Consider the parabola y = 9x - x^2.
    (a) Find the slope of the tangent line to the parabola at the point (1, 8).
    _________
    (b) Find an equation of the tangent line in part (a).
    y=_________
     
    #42
  43. Steady Eddy

    Steady Eddy Hall of Fame

    Joined:
    Jul 19, 2007
    Messages:
    2,545
    Location:
    Arizona
    I don't understand, the author's name is Stewart Multivariable?

    These are easier than they sound. Be sure to know them for a test 'cause they're likely to ask it, and it's easy points if you know it. Let's write it as a function. f(x) = 9x - x^2
    so f'(x) = 9 - 2x and f'(1) = 7.
    So going back to algebra, we have x = 1 y = 8 m = 7. Use y = mx + b, or y - y1 = m(x - x1) to find the equation. You should get y = 7x + 1. Is that right? Well, the slope is obviously 7 when x is 1. And if x = 1, then y = 8. So it works on both counts. So..
    (a) slope of the tangent line equals 7
    (b) y = 7x + 1
     
    #43
  44. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    Thanks man. It's called James Stewart : Multivariable Calculus 6th edition. Good book?

    Yeah that was really simple, I'll be able to remember how to do that.

    How about this one?
    If the tangent line to y = f (x) at (4,3) passes through the point (0,2), find the following.
    (a) f (4)

    The line would be y'=1/4x+ 2 I know, but where from there?

    And this....

    Consider the following curve.
    y = 8 + 5x2 - 2x3.

    (a) Find the slope of the tangent to the curve at the point where x = a.
    m =

    (b) Find the equation of the tangent line at the point (1, 11).
    f(x) =

    (c) Find the equation of the tangent line at the point (2, 12).
    g(x) =

    Thanks so much for the help.
     
    #44
  45. Steady Eddy

    Steady Eddy Hall of Fame

    Joined:
    Jul 19, 2007
    Messages:
    2,545
    Location:
    Arizona
    That's the one I see in Arizona everywhere. I think it's a typical, 3 semester, calculus book. I think the best book I've seen is, "How to Ace Calculus; A Streetwise Guide". It's easier to follow, and it can predict what question you can expect of the tests!

    For the first one, you don't have to do any math if you're alert. We're given the point (4,3), then they ask us to find f(4). It's not necessary to find the function, because f(4) has to be 3.

    The next one: y = 8 + 5x^2 - 2x^3
    y' = 10x - 6x^2
    x = a
    y' = 10a - 6a^2
    (that's the answer to part a)

    if a = 1, then y' = 4, so m = 4, x = 1, y = 11, gives us y = 4x + 7
    (that's b)
    if a = -1 then y' = -4, so m = -1, y = 12, x = 2, gives us y = -4x + 20
    (that's c)
     
    #45
  46. diredesire

    diredesire Super Moderator

    Joined:
    Mar 16, 2004
    Messages:
    6,720
    Bamboo, please send me an e-mail :)
     
    #46
  47. dak95_00

    dak95_00 Professional

    Joined:
    Sep 9, 2007
    Messages:
    864
    Location:
    Columbus, OH
    Steady,

    What is your background?

    I am finding this thread humorous as I am a high school math teacher who is teaching Calculus for the first time in a few years.

    Yahoo Answers is also very good to ask questions and get answers as there are many math nerds (no offense intended) there to answer many questions.

    You might also try http://www.calc101.com too.
     
    #47
  48. Morpheus

    Morpheus Professional

    Joined:
    Feb 19, 2004
    Messages:
    1,136
    Correct, but what was the question again?
     
    #48
  49. Steady Eddy

    Steady Eddy Hall of Fame

    Joined:
    Jul 19, 2007
    Messages:
    2,545
    Location:
    Arizona
    Humorous? Why? (I'm a retired H.S. math teacher)

    But he's asking me, not the other way around. Be like someone in a car saying, "Where's Glendale?" Me, "I don't know." Them, "Well, go to the gas station, they'll give you directions." I'm the one supplying answers, not asking questions.
     
    #49
  50. bamboo711

    bamboo711 Rookie

    Joined:
    Jul 4, 2005
    Messages:
    127
    Thanks Steady Eddy, your answers and explanations are wonderful.

    I really don't find this book that good at all to learn from, and I wish they had used a different book that would have taught in a different manner? I mean I consider myself pretty good at math (700 SAT 32 ACT on Math sections and I find the book not too useful.

    You might find this thread humorous, but Steady Eddy is pretty sick at explaining his method and answers which I appreciate. Pretty much I don't intend on using math in any sense in the future, and I have to get through this class and make at least like a 3.5 in it, so Eddy is making it that much smoother. Thanks man.

    More questions to come soon if that's okay.
     
    #50

Share This Page