To solve the formula for the eccentricity [tex]\( e \)[/tex] of an orbit for the length of the semi-major axis [tex]\( a \)[/tex], we start with the given equation:
[tex]\[
e = \sqrt{1 - \frac{b^2}{a^2}}
\][/tex]
First, square both sides to eliminate the square root:
[tex]\[
e^2 = 1 - \frac{b^2}{a^2}
\][/tex]
Next, isolate the fraction by subtracting [tex]\( e^2 \)[/tex] from both sides:
[tex]\[
\frac{b^2}{a^2} = 1 - e^2
\][/tex]
Then, solve for [tex]\( a^2 \)[/tex] by rearranging the equation:
[tex]\[
a^2 = \frac{b^2}{1 - e^2}
\][/tex]
Finally, take the square root of both sides to solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{b}{\sqrt{1 - e^2}}
\][/tex]
Thus, the correct equation for the length of the semi-major axis is equation [tex]\( D \)[/tex].
When the length of the major axis [tex]\( b \)[/tex] is 4 inches and the eccentricity [tex]\( e \)[/tex] is [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[
a = \frac{4}{\sqrt{1 - \left(\frac{1}{2}\right)^2}}
\][/tex]
Calculate the eccentricity squared:
[tex]\[
e^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
\][/tex]
Substitute [tex]\( e^2 \)[/tex] into the equation:
[tex]\[
a = \frac{4}{\sqrt{1 - \frac{1}{4}}} = \frac{4}{\sqrt{\frac{3}{4}}}
\][/tex]
Simplify the expression under the square root:
[tex]\[
a = \frac{4}{\frac{\sqrt{3}}{2}} = \frac{4 \times 2}{\sqrt{3}} = \frac{8}{\sqrt{3}}
\][/tex]
Rationalize the denominator:
[tex]\[
a = \frac{8 \sqrt{3}}{3} \approx 4.62
\][/tex]
So, the length of the semi-major axis is approximately 4.62 inches.
### Final answers:
The correct equation for the length of the semi-major axis is equation [tex]\( \boxed{D} \)[/tex].
When the length of the major axis is 4 inches and the eccentricity of the orbit is [tex]\( \frac{1}{2} \)[/tex], the length of the semi-major axis is [tex]\( \boxed{4.62} \)[/tex] inches (rounded to the nearest hundredth).
