# Thread: Area Between 2 Curves and Finding Tangent Parabola Translation Constant

1. ## 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.

2. ## Re: Area Between 2 Curves and Finding Tangent Parabola Translation Constant

Originally Posted by kwikness
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.

3. ## Re: Area Between 2 Curves and Finding Tangent Parabola Translation Constant

Originally Posted by kwikness
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 $\displaystyle y_1$ and $\displaystyle y_2$ are tangent then they must have the same gradient:

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

and

$\displaystyle y_2' = 0.16x$

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

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

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

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