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Math Help - Direction

  1. #1
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    Direction

    The planned path of a UFO is defined by the equation

    y=0.25x^4 + 0.6667x^3 for x > 0

    units are in km

    (a) Find the direction of motion when the x-value is 2 and 3

    (b) Find a point on the UFOs path where it is inclined at 45 degrees to the positive x-axis.

    (c) Are there any other points that are similar to (b).

    If someone could help i would be very greatful
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  2. #2
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    You're given a distance formula.

    The derivative of distance is velocity, so for (a) derive the equation and find the sign of the velocity equation at those points.

    For part b/c, just picture a 45-45-90 triangle, just for reference. Each leg of the triangle has a length of \frac{\sqrt{2}}{2}*x, but in this calculation x doesn't matter.

    So if you draw a triangle which has a left endpoint connected to the origin, then the hypotenuse becomes a line with a slope 45 degrees with respect to the x axis.

    \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1

    So for (b) you just need to find where on x > 0 the slope of y = 1. Which should be easy assuming you have a basic understanding of these concepts.
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  3. #3
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    Quote Originally Posted by thekiterunner View Post
    The planned path of a UFO is defined by the equation

    y=0.25x^4 + 0.6667x^3 for x > 0

    units are in km

    (a) Find the direction of motion when the x-value is 2 and 3
    Is this one or two questions. If you mean "Find the directions of motion when x= 2 and x= 3" then the angle the direction makes with the x-axis is the arctangent of the derivative. And with y= \frac{1}{4}x^4+ \frac{2}{3}x^3, the derivative is y'= x^3+ 2x^2. At 2 and 3 the derivatives are 16 and 45 respectively.

    [quote](b) Find a point on the UFOs path where it is inclined at 45 degrees to the positive x-axis.[quote]
    tan(45)= 1 so you want to solve y'= x^3+ 2x^2= 1.

    (c) Are there any other points that are similar to (b).
    y'= x^3+ 2x^2= 1, being cubic, may have three solutions. I guess that is what is meant by "similar to (b)".

    If someone could help i would be very greatful
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  4. #4
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    So that means solving for x^3 +2x^2 = 1

    is X^3 + 2x^2 -1
    (x+1)(x^2+x-1)

    so x = -1
    x=0.618
    x=-1.618

    Another way of say it is
    x^3 +2x^2=x

    All the points where X^3 +2x^2 crosses the line function x

    then I get

    x = 0.414
    x=-2.414
    x=0

    Which one of the answers is correct?
    Last edited by thekiterunner; October 27th 2009 at 01:34 PM.
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