Solve the problem.
-Given that the polar equation $\mathrm { r } = \frac { \mathrm { a } \left( 1 - \mathrm { e } ^ { 2 } \right) } { 1 + \mathrm { e } \cos \theta }$ models the orbits of the planets, estimate the closest approach of Pluto's orbit, for which $\mathrm { a } = 39.4$ and $\mathrm { e } = 0.249$, to the sun.
A) About 50 astronomical units
B) About 74 astronomical units
C) About 39 astronomical units
D) About 30 astronomical units

Question

Quizplus · Accepted Answer

The closest approach, or perihelion, of an orbit in a polar equation of the form $ r = \frac { a ( 1 - e ^ { 2 } ) } { 1 + e \cos \theta }$ occurs when $\theta = 0$, which maximizes the denominator. Substituting $\theta = 0$, $a = 39.4$, and $e = 0.249$ into the equation gives the closest approach distance: $ r = \frac { 39.4 ( 1 - 0.249 ^ { 2 } ) } { 1 + 0.249 }$, which calculates to approximately 29.7 astronomical units, rounded to about 30 astronomical units.

Solve the Problem r=a(1−e2)1+ecos⁡θ\mathrm { r } = \frac { \mathrm { a } \left( 1 - \mathrm { e } ^ { 2 } \right) } { 1 + \mathrm { e } \cos \theta }r=1+ecosθa(1−e2)​

Solve the Problem $\mathrm { r } = \frac { \mathrm { a } \left( 1 - \mathrm { e } ^ { 2 } \right) } { 1 + \mathrm { e } \cos \theta }$