Step 1: Understanding the Concept:
The eccentricity of an ellipse measures its deviation from being circular. To find it, we first need to convert the given equation into the standard form of an ellipse and identify the major and minor axes.
Step 2: Key Formula or Approach:
The standard equation of an ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ or $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$, where $a$ is the semi-major axis and $b$ is the semi-minor axis ($a>b$). The eccentricity $e$ is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Step 3: Detailed Explanation:
The given equation is $16x^2 + 7y^2 = 112$.
To convert it to the standard form, divide the entire equation by 112:
\[ \frac{16x^2}{112} + \frac{7y^2}{112} = \frac{112}{112} \]
\[ \frac{x^2}{7} + \frac{y^2}{16} = 1 \]
By comparing this with the standard form, we can identify $a^2$ and $b^2$. Since $16>7$, the major axis is along the y-axis.
\[ a^2 = 16 \implies a = 4 \]
\[ b^2 = 7 \implies b = \sqrt{7} \]
Now, we can calculate the eccentricity using the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1 - \frac{7}{16}} \]
\[ e = \sqrt{\frac{16 - 7}{16}} = \sqrt{\frac{9}{16}} \]
\[ e = \frac{3}{4} \]
Step 4: Final Answer:
The eccentricity of the given ellipse is $\frac{3}{4}$. Therefore, option (D) is correct. Note that eccentricity for an ellipse is always between 0 and 1. Option (A) is greater than 1, so it can be immediately ruled out.