Question

Halley's comet has an elliptical orbit with the sun at one focus. Its orbit shown below is given approximately by 10.71 r - 1 + 0.883 sin θIn the formula, r is measured in astronomical units. (One astronomical unit is the average distance from Earth to the sun, approximately 93 million miles.) Find the distance from Halley's comet to the sun at its greatest distance from the sun. Round to the nearest hundredth of an astronomical unit and the nearest million miles.


A. 12.13 astronomical units; 1128 million miles

B. 91.54 astronomical units; 8513 million miles

C. 5.69 astronomical units; 529 million miles

D. 6.06 astronomical units; 564 million miles

Answer 1

The formula of an elliptical orbit is given by

$r = \frac{A}{1+Bsin\theta}$

Assuming the given expression that was wrongly typed and whose true function is

$r = \frac{10.71}{1+0.883sin\theta}$

We could start by deducing that the greatest distance from the sun would be given at the angle

$\theta = \frac{3\pi}{2}$

For that value the value of $sin\theta=-1$

$r = \frac{10.71}{1+0.883(-1)}$

$r = 91.538 AU$

That is equal to

$r = 91.54Au* (\frac{93*10^6milles}{1AU})$

$r = 8513$ million miles

Therefore the correct option is B.