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Math Help - Some Algebra 2 level problems I am having a lot of trouble solving=(

  1. #1
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    Some Algebra 2 level problems I am having a lot of trouble solving=(

    Can someone dedicated please answer these questions and take a brief moment to explain? Please show work, thanks a lot!!!!

    1. Devon tosses a horseshoe at a stake 30 feet away. The horseshoe lands no more than 3 feet from the stake. (a) Write an absolute value inequality that represents the range of distances that the horseshoe travels. (b) Solve the inequality.

    2. Suppose the altitude of a rising hot-air balloon is given by (^indicates to a power) h = 0.04 t^2 + 2t, where “t” is the time in seconds after the balloon leaves the ground. How long will it take for the balloon to reach an altitude of 200 feet? "h" is height.

    Question #3. 3b[sqrt(27a^5b)] + 2a[sqrt(3a^3b^3)] simplify (rationalize denominator if needed) - sqrt means square root

    4. 3/[sqrt(5) - 2] simplify (rationalize denominator if needed) - sqrt means square root

    5. solve and check for extraneous roots sqrt(x + 4) = sqrt(x) - 2
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    What have you got so far?

    The first one I would draw a diagram to help you conceptualize what the question is asking. Do you know what an absolute value is?

    The second one you're just plugging in for H and solving for time (you will get an extraneous solution; see if you can find which one it is).

    The third one asks you to understand the properties of radicals such as:

    \sqrt{a^3}=\sqrt{a^{2}a}=a\sqrt{a}

    On the fourth one, you will have to multiple by a conjugate. Are you familiar with conjugates. I would imagine so if they are asking you to get the radical out of the denominated. The conjugate of a+b is a-b.

    On the fifth, simply a matter of rearranging the equation, expanding the square and solving for zero.
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    1. Devon tosses a horseshoe at a stake 30 feet away. The horseshoe lands no more than 3 feet from the stake. (a) Write an absolute value inequality that represents the range of distances that the horseshoe travels. (b) Solve the inequality.

    Yes, absolute value is the distance of x from 0. This one I have no clue how to solve.

    2. Suppose the altitude of a rising hot-air balloon is given by (^indicates to a power) h = 0.04 t^2 + 2t, where t is the time in seconds after the balloon leaves the ground. How long will it take for the balloon to reach an altitude of 200 feet? "h" is height.

    200 = 0.04(t^2) + 2t - I have to solve this one by factoring.. Where do I begin?

    Question #3.

    3b[sqrt(27a^5b)] + 2a[sqrt(3a^3b^3)] simplify (rationalize denominator if needed) - sqrt means square root

    -No clue-

    4. 3/[sqrt(5) - 2] simplify (rationalize denominator if needed) - sqrt means square root

    I am not familiar with conjugates.

    5. solve and check for extraneous roots sqrt(x + 4) = sqrt(x) - 2

    How would I do that?

    If someone would have the time to work each one out completely and write out how everything is done, it would be highly appreciated.
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  4. #4
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    Yes, absolute value is the distance of x from 0. This one I have no clue how to solve.
    You're on the right track. Absolute value is the NUMERICAL value of a number; meaning |-2| and |3| are the same - 3. Think of of this way, if I said I gave you 3 apples (3) and took away 3 apples later (-3), how many apples overall are we talking about.

    The same applies in this situation (although its a silly word problem). Like I said, draw a diagram and label it with what we know:

    The distance from the man to the horse shoe spike: 30 feet
    How close can he get to the horse shoe spike: No closer than 3 feet
    How far can the spike travel: X

    Where do I begin?
    Rearrange the equation so that it equals zero. You can certain factor out a 2 but that wont help you. Have you learned the quadratic formula? They want you to use it here, and look for any answers that dont make sense.

    -No clue-
    Yes you do. Even if you don't think you do. You're used to seeing:

    \sqrt{4}=2

    But isn't that just, \sqrt{2*2}=2? Or more generally \sqrt{a*a}=a? What about \sqrt{8}=\sqrt{2*2*2}=2\sqrt{2}. Do you see how I was able to pull out the 2*2? Understanding that is key to understanding what they're saying that the number that is that multiplied by itself and equals 8 is 2\sqrt{2} . In the problem:

    3b\sqrt{27a^{5}b}+2a\sqrt{3a^{3}b^{3}} do the exact same thing. Write ALL a's and b's out: a^{5}=a*a*a*a*a. Do the same thing to the 27 (factor the 27, and see what you get).

    I am not familiar with conjugates.
    Is this the problem:

    \frac{3}{\sqrt{5}-2}. If so they want you to rationalize the denominator, which means you are meant to get the radical (in this case the square root sign, but we also have cubed-roots, fourth roots, fifth roots etc) out of the denominator and into the numerator. However how can we go about doing that? Have a look at the definition of a conjugate here and see if it makes sense: Conjugate (algebra)

    How would I do that?
    Rearrange the problem:

    \sqrt{x+4}=\sqrt{x}-2 \Rightarrow \sqrt{x+4}+2=\sqrt{x} \Rightarrow (\sqrt{x+4}+2)^{2}=x

    Can you take it from there?
    Last edited by ANDS!; August 3rd 2009 at 11:12 AM.
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  5. #5
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    Quote Originally Posted by bobbyboy1111 View Post

    1. Devon tosses a horseshoe at a stake 30 feet away. The horseshoe lands no more than 3 feet from the stake. (a) Write an absolute value inequality that represents the range of distances that the horseshoe travels. (b) Solve the inequality.

    Yes, absolute value is the distance of x from 0. This one I have no clue how to solve.

    2. Suppose the altitude of a rising hot-air balloon is given by (^indicates to a power) h = 0.04 t^2 + 2t, where t is the time in seconds after the balloon leaves the ground. How long will it take for the balloon to reach an altitude of 200 feet? "h" is height.

    200 = 0.04(t^2) + 2t - I have to solve this one by factoring.. Where do I begin?

    Question #3.

    3b[sqrt(27a^5b)] + 2a[sqrt(3a^3b^3)] simplify (rationalize denominator if needed) - sqrt means square root

    -No clue-

    4. 3/[sqrt(5) - 2] simplify (rationalize denominator if needed) - sqrt means square root

    I am not familiar with conjugates.

    5. solve and check for extraneous roots sqrt(x + 4) = sqrt(x) - 2

    How would I do that?

    If someone would have the time to work each one out completely and write out how everything is done, it would be highly appreciated.
    A. ANDS: I think that's a typo in Q1: |-2| = 2

    0. Nobody will answer these for you in full without interaction or willingness on your part

    1. The absolute value can be used to represent the distance from the stake

    2. 0.04 = \frac{1}{25} - if you feel lazy use the quadratic formula and solve that way. Remember that t > 0

    3. See Above post

    4. If you're unfamiliar with conjugates think in terms of the difference of two squares: (a+b)(a-b) = a^2 - b^2. Those two terms on the left happen to be conjugates and the squared terms means no square roots.

    5. See 3
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    Here is what I got for 3 so far, but there is no denominator to rationalize:

    (3b)(3a^2SQRT(3ab)) + (2a)(abSQRT(3ab))


    (3b * 3a^2SQRT(3ab)) + (2a * abSQRT(3ab))


    (9a^2bSQRT(3ab)) + (2a^2bSQRT(3ab))


    9a^2bSQRT(3ab) + 2a^2bSQRT(3ab)


    11a^2bSQRT(3ab)

    Is this correct? I will post more as I figure it out.
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    That is it.
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    Question #1 - The spike can travel -3 away from 30 and 3 from 30. We don't know on which side it landed so our range would be represented by -
    |h - 3| >= 3, correct?
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    Quote Originally Posted by bobbyboy1111 View Post
    Question #1 - The spike can travel -3 away from 30 and 3 from 30. We don't know on which side it landed so our range would be represented by -
    |h - 3| >= 3, correct?
    The reasoning is correct (without the minus sign on the 3, because remember we are talking about ABSOLUTE distance), but the equation is not, unless you misplaced a zero. If you did not make a typo in your equation:

    Imagine the distance to the spike is on the X-axis. It is 30 feet from the mans origin. The distance that he throws the horse shoe subtracted from the distance to the spike can be no more than 3 feet yes? How would I find the absolute distance in difference from one point to another than can be on either side of my original point (the spike)?
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    ANDS!, we will continue tomorrow, ok? Thanks a lot for the help so far Roughly what time/s will you be online? U.S. times?


    Thanks again, Bobbyboy1111
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    2. Suppose the altitude of a rising hot-air balloon is given by (^indicates to a power) h = 0.04 t^2 + 2t, where t is the time in seconds after the balloon leaves the ground. How long will it take for the balloon to reach an altitude of 200 feet? "h" is height.

    How would I set this to 0 in order to solve by factoring? Like this:

    (0.04)t^2 + 2t - 200 ????

    What do I do with the 0.04?
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    4. 3/[SQRT(5) - 2]

    3/[SQRT(5) - 2] * SQRT(5) + 2/SQRT(5) + 2

    3(SQRT(5) + 2)/(SQRT5)^2 -(-2)^2

    3(SQRT(5) + 2)/5 - (8)

    - SQRT(5) - 2

    Is this correct? How would I rationalize this? Can it be done?
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    Quote Originally Posted by bobbyboy1111 View Post
    2. Suppose the altitude of a rising hot-air balloon is given by (^indicates to a power) h = 0.04 t^2 + 2t, where “t” is the time in seconds after the balloon leaves the ground. How long will it take for the balloon to reach an altitude of 200 feet? "h" is height.

    How would I set this to 0 in order to solve by factoring? Like this:

    (0.04)t^2 + 2t - 200 ????

    What do I do with the 0.04?
    If you don't like the decimal, you could multiply both sides by 25 (because 0.04 times 25 equals 1):
    0.04t^2 + 2t - 200 = 0

    t^2 + 50t - 5000 = 0

    Looks like this is factorable...
    (t + 100)(t - 50) = 0

    I'll let you take it from here.


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    Quote Originally Posted by ANDS! View Post
    The reasoning is correct (without the minus sign on the 3, because remember we are talking about ABSOLUTE distance), but the equation is not, unless you misplaced a zero. If you did not make a typo in your equation:

    Imagine the distance to the spike is on the X-axis. It is 30 feet from the mans origin. The distance that he throws the horse shoe subtracted from the distance to the spike can be no more than 3 feet yes? How would I find the absolute distance in difference from one point to another than can be on either side of my original point (the spike)?
    How about this: | x | < 3, or 3 < x < 3

    My original answer was this, but it turned out to be incorrect:

    |h - 3| >= 30

    h - 3 <= -30 or h - 3 >= 30


    h <= -27 or h >= 33
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    Quote Originally Posted by bobbyboy1111 View Post
    4. 3/[SQRT(5) - 2]

    3/[SQRT(5) - 2] * SQRT(5) + 2/SQRT(5) + 2

    3(SQRT(5) + 2)/(SQRT5)^2 -(-2)^2

    3(SQRT(5) + 2)/5 - (8)
    Whoa, where did you get the 8?
    (\sqrt{5} - 2)(\sqrt{5} + 2) = 5 - 4 = 1

    I'm getting 3\sqrt{5} + 6.


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