Question

Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection. Give your answers in degrees, rounding to one decimal place. Enter your answers as a comma-separated list.) y = 8x2, y = 8x3

Answer 1

Answer:

1.190390345 degrees.

Explanation:

As stated in the problem that the angle between the two curves is also the angle between slopes of the two cures at the point of intersection, which is 1 (when we set two equation equal and solve for x).

We know, if you don't  you could verify it for yourself and it will be a nice mathematical exercise for you, that when there are two lines with slopes $m_1$ and $m_2$ then the angle between them and their two slopes has following relationship.

$tan(\alpha ) = |\frac{m_2-m_1}{1+m_1m_2}|$   where $\alpha$ is angle between the two lines.

As it is clear that we can easily get the angle between the two curves if we know slopes at that point, which you will see in second is very straight forward to calculate.

Slopes are simply derivatives evaluated at the point of intersection and the two derivatives are $D_1$ 16x and $D_2 =24x^2$, substituting x =1 we get, $m_1 = 16$ and $m_2 = 24$.

Now put $m_1$ and $m_2$ in this relationship, rearrange and solve for angle, it will come out to be

$\alpha$ = 1.190390345 degrees.

that is our angle that we want to know between the two cures at the point of their intersection.