Question 1075331
<pre>
{{{"y''" = 12x^2 &#8722; 8}}}

{{{"d(y')"/(dx)=12x^2-8}}}

{{{"d(y')"=12x^2dx-8dx}}}

{{{int("d(y')")}}}{{{""=""}}}{{{int(12x^2dx)}}}{{{""-""}}}{{{int(8dx)}}}

{{{"y'"}}}{{{""=""}}}{{{12int(x^2dx)}}}{{{""-""}}}{{{8int(dx)}}}

{{{"y'"}}}{{{""=""}}}{{{12(x^3/3)}}}{{{""-""}}}{{{8x}}}{{{""+""}}}{{{C}}}

{{{"y'"}}}{{{""=""}}}{{{4x^3}}}{{{""-""}}}{{{8x}}}{{{""+""}}}{{{C}}}

The slope of the line  x + 224y &#8722; 448 = 0 is found by taking the derivative
implicitly             1 + 224y' - 0  = 0
                                224y' = -1
                                   y' = {{{-1/224}}} <--- slope

Since the tangent to the curve at (4,192) has slope which is perpendicular
to that, it has slope which is its negative reciprocal or +224.

So we set y' = 224, and x = 4 in


{{{224}}}{{{""=""}}}{{{4(4)^3}}}{{{""-""}}}{{{8(4)}}}{{{""+""}}}{{{C}}}
{{{224}}}{{{""=""}}}{{{4*64}}}{{{""-""}}}{{{32}}}{{{""+""}}}{{{C}}}
{{{224}}}{{{""=""}}}{{{256}}}{{{""-""}}}{{{32}}}{{{""+""}}}{{{C}}}
{{{224}}}{{{""=""}}}{{{224}}}{{{""+""}}}{{{C}}}
{{{0}}}{{{""=""}}}{{{C}}}

Substitution C=0 in

{{{"y'"}}}{{{""=""}}}{{{4x^3}}}{{{""-""}}}{{{8x}}}{{{""+""}}}{{{C}}}

{{{"y'"}}}{{{""=""}}}{{{4x^3}}}{{{""-""}}}{{{8x}}}

{{{(dy)/(dx)}}}{{{""=""}}}{{{4x^3}}}{{{""-""}}}{{{8x}}}

{{{dy}}}{{{""=""}}}{{{4x^3dx}}}{{{""-""}}}{{{8xdx}}}

{{{int(dy)}}}{{{""=""}}}{{{int(4x^3dx)}}}{{{""-""}}}{{{int(8xdx)}}}

{{{y}}}{{{""=""}}}{{{4int(x^3dx)}}}{{{""-""}}}{{{8int(xdx)}}}

{{{y}}}{{{""=""}}}{{{4(x^4/4)}}}{{{""-""}}}{{{8(x^2/2)}}}

{{{y}}}{{{""=""}}}{{{4(x^4/4)}}}{{{""-""}}}{{{8(x^2/2)}}}

{{{y}}}{{{""=""}}}{{{x^4}}}{{{""-""}}}{{{4x^2)}}}{{{""+""}}}{{{C}}}

Since the curve passes through the point (4, 192) we substitute
x=4 and y=192

{{{192}}}{{{""=""}}}{{{4^4}}}{{{""-""}}}{{{4(4)^2)}}}{{{""+""}}}{{{C}}}

{{{192}}}{{{""=""}}}{{{256}}}{{{""-""}}}{{{64)}}}{{{""+""}}}{{{C}}}

{{{192}}}{{{""=""}}}{{{192}}}{{{""+""}}}{{{C}}}

{{{0}}}{{{""=""}}}{{{C}}}

So to find the equation we substitute C=0 in

{{{y}}}{{{""=""}}}{{{x^4}}}{{{""-""}}}{{{4x^2)}}}{{{""+""}}}{{{C}}}

{{{y}}}{{{""=""}}}{{{x^4}}}{{{""-""}}}{{{4x^2)}}}{{{""+""}}}{{{0}}}

{{{y}}}{{{""=""}}}{{{x^4}}}{{{""-""}}}{{{4x^2)}}}

Edwin</pre>