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Math Help - I need some major help reviewing please

  1. #1
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    I need some major help reviewing please

    My calculus teacher gave us a review sheet today that has several problems on it that are like what is going to be on our test that we are having Thursday. The problem is that he will be out of town until Thursday morning, so I can't go and ask him for help. I will post two of the problems that I am having issues with here. Please help me!!

    1) Consider the polar equation r=24/5+sinx
    a)Transform the polar equation into rectangular coordinates.
    b) Find the foci of the ellipse formed

    2) a) If A is the point (0,0,1) find a point B so that the vector AB has length 6 and is parallel to <-1,4,1>

    b) find an equation of the sphere for which the line segment from A to C (4,-6,9) is a diameter

    Thanks in advance!!
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    Re: I need some major help reviewing please

    Quote Originally Posted by blackflag10 View Post
    My calculus teacher gave us a review sheet today that has several problems on it that are like what is going to be on our test that we are having Thursday. The problem is that he will be out of town until Thursday morning, so I can't go and ask him for help. I will post two of the problems that I am having issues with here. Please help me!!

    1) Consider the polar equation r=24/5+sinx
    a)Transform the polar equation into rectangular coordinates.
    b) Find the foci of the ellipse formed

    2) a) If A is the point (0,0,1) find a point B so that the vector AB has length 6 and is parallel to <-1,4,1>

    b) find an equation of the sphere for which the line segment from A to C (4,-6,9) is a diameter

    Thanks in advance!!
    Is this \displaystyle \begin{align*} r = \frac{24}{5} + \sin{x}  \end{align*} or \displaystyle \begin{align*} \frac{24}{5\sin{x}} \end{align*} ?
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    Re: I need some major help reviewing please

    It is r= 24/(5+sinx)
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    Re: I need some major help reviewing please

    Quote Originally Posted by blackflag10 View Post
    It is r= 24/(5+sinx)
    You should know that \displaystyle \begin{align*} x = r\cos{\theta}, y = r\sin{\theta} \end{align*} and \displaystyle \begin{align*} x^2 + y^2 = r^2 \end{align*}

    Therefore

    \displaystyle \begin{align*} r &= \frac{24}{5 + \sin{x}} \\ r &= \frac{24}{5 + \frac{y}{r}} \\ r\left(5 + \frac{y}{r}\right) &= 24 \\ 5r + y &= 24 \\ 5\sqrt{x^2 + y^2} &=  24 - y \\ 25\left(x^2 + y^2\right) &= \left(24 - y\right)^2 \\ 25x^2 + 25y^2 &= y^2 - 48y + 576 \\ 25x^2 + 24y^2 + 48y &= 576 \\ 25x^2 + 24\left(y^2 + 2y\right) &= 576 \\ 25x^2 + 24\left(y^2 + 2y + 1^2 - 1^2\right) &= 576 \\ 25x^2 + 24\left[\left(y + 1\right)^2 - 1\right] &= 576 \\ 25x^2 + 24\left(y + 1\right)^2 - 24 &= 576 \\ 25x^2 + 24\left(y + 1\right)^2 &= 600 \\ \frac{x^2}{24} + \frac{\left(y + 1\right)^2}{25} &= 1  \end{align*}

    You should be able to find the foci now.
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    Re: I need some major help reviewing please

    Quote Originally Posted by Prove It View Post
    You should know that \displaystyle \begin{align*} x = r\cos{\theta}, y = r\sin{\theta} \end{align*} and \displaystyle \begin{align*} x^2 + y^2 = r^2 \end{align*}

    Therefore

    \displaystyle \begin{align*} r &= \frac{24}{5 + \sin{x}} \\ r &= \frac{24}{5 + \frac{y}{r}} \\ r\left(5 + \frac{y}{r}\right) &= 24 \\ 5r + y &= 24 \\ 5\sqrt{x^2 + y^2} &=  24 - y \\ 25\left(x^2 + y^2\right) &= \left(24 - y\right)^2 \\ 25x^2 + 25y^2 &= y^2 - 48y + 576 \\ 25x^2 + 24y^2 + 48y &= 576 \\ 25x^2 + 24\left(y^2 + 2y\right) &= 576 \\ 25x^2 + 24\left(y^2 + 2y + 1^2 - 1^2\right) &= 576 \\ 25x^2 + 24\left[\left(y + 1\right)^2 - 1\right] &= 576 \\ 25x^2 + 24\left(y + 1\right)^2 - 24 &= 576 \\ 25x^2 + 24\left(y + 1\right)^2 &= 600 \\ \frac{x^2}{24} + \frac{\left(y + 1\right)^2}{25} &= 1  \end{align*}

    You should be able to find the foci now.
    Thank you so much!!!

    If someone could help me out with the second question, I would be very grateful!
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    Re: I need some major help reviewing please

    Quote Originally Posted by blackflag10 View Post
    My calculus teacher gave us a review sheet today that has several problems on it that are like what is going to be on our test that we are having Thursday. The problem is that he will be out of town until Thursday morning, so I can't go and ask him for help. I will post two of the problems that I am having issues with here. Please help me!!

    1) Consider the polar equation r=24/5+sinx
    a)Transform the polar equation into rectangular coordinates.
    b) Find the foci of the ellipse formed

    2) a) If A is the point (0,0,1) find a point B so that the vector AB has length 6 and is parallel to <-1,4,1>

    b) find an equation of the sphere for which the line segment from A to C (4,-6,9) is a diameter

    Thanks in advance!!
    The vector <-1, 4, 1> represents the vector from the point (0, 0, 0) to (-1, 4, 1).

    You wish the vector to be parallel to this and to begin at (0, 0, 1), so to be translated one unit up in the z axis.

    So a vector in the direction of <-1, 4, 1> starting at (0, 0, 1) is <-1, 4, 2>.

    Now you want this vector to have a length of 6, so if we divide it by its own length, it will be a unit vector, and then multiplying by 6 will give it a length of 6.
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