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Math Help - Area Between 2 Curves and Finding Tangent Parabola Translation Constant

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    Area Between 2 Curves and Finding Tangent Parabola Translation Constant

    Hi guys, I'm having trouble with this word problem. I can figure out how to find the area (part b) once I determine what interval I'm integrating over, but I'm not sure about how to find the interval, or find k where the parabola is tangent to y1.

    Thanks.
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    Senior Member abhishekkgp's Avatar
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    Re: Area Between 2 Curves and Finding Tangent Parabola Translation Constant

    Quote Originally Posted by kwikness View Post
    Hi guys, I'm having trouble with this word problem. I can figure out how to find the area (part b) once I determine what interval I'm integrating over, but I'm not sure about how to find the interval, or find k where the parabola is tangent to y1.

    Thanks.
    let y=mx be tangent to the parabola. let the equation of the parabola be y=f(x). then the line cuts the parabola at only one point, that is, mx=f(x) has only one root. just do the discriminant=0 thingy.
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    Re: Area Between 2 Curves and Finding Tangent Parabola Translation Constant

    Quote Originally Posted by kwikness View Post
    Hi guys, I'm having trouble with this word problem. I can figure out how to find the area (part b) once I determine what interval I'm integrating over, but I'm not sure about how to find the interval, or find k where the parabola is tangent to y1.

    Thanks.
    1. If the curves of y_1 and y_2 are tangent then they must have the same gradient:

    |y_1'| = \left|\dfrac{dy_1}{dx} \right| = 1

    and

    y_2' = 0.16x

    2. Taking the right branch of y_1 you have to solve for x:

    0.16x = 1~\implies~x= \frac{25}4

    3. Therefore the tangent point is T\left( \frac{25}4 , \frac{25}4 \right)

    4. Now determine the interval of the integration part (I would use some symmetry properties of the curves to simplify the calculations)
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