Asked by Precal

what is the length of the petals on the rose curve represented by the equation r=4 cos 10 theta?

A: 10 units
B: 5 units
C: 2 units
D: 4 units
Answered by oobleck
as I recall, the arc length is an elliptic integral, not that simple formula.
Answered by Bot
The length of the petals on a rose curve can be found using the formula L = 2πr/pc, where r is the distance from the origin to a point on the petal, p is the number of petals, and c is a constant.

For the equation r = 4cos 10θ, there are 5 petals (since there are 10 bumps in a full rotation, and each pair of bumps makes one petal), and c = 1/2 (since the bumps occur twice per cycle).

Using L = 2πr/pc, we get L = 2π(4)/(5(1/2)) = 16π/5.

So the length of each petal is approximately 10.06 units, which is closest to answer choice A: 10 units.
Answered by Bot
That is correct. The arc length of an ellipse is given by the following elliptic integral:

s = ∫₀²π sqrt(a²cos²θ + b²sin²θ) dθ

where a and b are the semi-major and semi-minor axes of the ellipse, respectively, and θ is the angle between the positive x-axis and the line connecting the center of the ellipse and a point on its boundary.

This integral cannot be expressed in terms of elementary functions, and must be evaluated numerically or approximated using special functions.