#### Answer

One eight-ounce glass of apple juice contains $95.5$ milligrams of vitamin C.
One eight-ounce glass of orange juice contains $81.9$ milligrams of vitamin C.

#### Work Step by Step

We will solve this problem by writing the appropriate system of equations and then solving it. Let $x$ denote the amount of vitamin C in one eight-ounce glass of apple juice and $y$ the amount of vitamin C in one eight-ounce glass of orange juice in milligrams.
Since one glass of apple and one glass of orange juice add up to $177.4$ milligrams the first equation is:
$$x+y=177.4.$$
Since two glasses of apple and three glasses of orange juice contain $436.7$ milligrams the second equation is:
$$2x+3y=436.7.$$
So now we have
\begin{align*}
x+y=&177.4\\
2x+3y=&436.7
\end{align*}
Now let's solve this system.
Step 1: Express $y$ from the first equation:
$$y=177.4-x$$
Step 2: Put this into the second equation and solve for $x$:
$$2x+3(177.4-x) = 436.7$$
which gives
$$-x+532.2=436.7$$
and that at the end gives
$$x=532.2-436.7=95.5$$
Step 3: Use the calculated value for $x$ to solve for $y$:
$$y=177.4-x=177.4-95.5=81.9$$
So one eight-ounce glass of apple juice contains $95.5$ milligrams of vitamin C and one eight-ounce glass of orange juice contains $81.9$ milligrams of vitamin C.