Halley's comet has an elliptical orbit with the sun at one focus. its orbit shown below is given approximately by r=(10.71)/(1+0.883sin\theta) in 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.

In this problem, we apply the concepts in calculus. when you want to find the maximum value of r, you find the derivative in terms of  θ. so, i would want to write the equation again so that we could understand it clearly



next, we differentiate r in terms of θ using the chain rule in calculus. for fractions in the form of u/v, the derived from would be (vdu -udv)v².

dr/dt = [(1+0.883sinθ)(0) - (10.71)(0+0.883cosθ)]/(1+0.883sinθ)² = 0
-10.71(0.883cosθ) = 0
cosθ =0/-10.71(0.883) = 0
θ = cos⁻¹(0)
θ = 90°

so, we substitute θ=90° to the original equation:

r = 10.71/(1+0.883 sin90°)
r = 5.69 astronomical units

then, we convert this to miles through the conversion 1 astronomical unit = 93 million miles:

5.69*(93/1) = 529 million miles