Math Problem Statement

Test:FINAL EXAM Question 2 of 20 This test: 20 point(s) possible This question: 1 point(s) possible

Skip to Main content Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 Question 16 Question 17 Question 18 Question 19 Question 20 Question content area top Part 1 The letters x and y represent rectangular coordinates. Write the given equation using polar coordinates left parenthesis r comma theta right parenthesis. x squared equals 4 y Question content area bottom Part 1 Select the equation in polar coordinates. A.r squared cosine theta minus 4 r sine theta equals 0 r squared cosine theta minus 4 r sine theta equals 0 B.4 r cosine theta minus r squared sine squared theta equals 0 4 r cosine theta minus r squared sine squared theta equals 0 C.4 r cosine theta minus sine theta equals 0 4 r cosine theta minus sine theta equals 0 D.r squared cosine squared theta minus 4 r sine theta equals 0 r squared cosine squared theta minus 4 r sine theta equals 0 Select the equation in polar coordinates. Time Remaining: 02:28:48

Solution

To convert the rectangular coordinate equation x2=4yx^2 = 4y into polar coordinates, we use the following transformations between rectangular coordinates (x,y)(x, y) and polar coordinates (r,θ)(r, \theta):

  • x=rcos⁡θx = r \cos \theta
  • y=rsin⁡θy = r \sin \theta

Substitute these into the equation x2=4yx^2 = 4y:

(rcos⁡θ)2=4(rsin⁡θ)(r \cos \theta)^2 = 4(r \sin \theta)

Expanding the left side gives:

r2cos⁡2θ=4rsin⁡θr^2 \cos^2 \theta = 4r \sin \theta

Rearranging the equation to bring all terms to one side, we get:

r2cos⁡2θ−4rsin⁡θ=0r^2 \cos^2 \theta - 4r \sin \theta = 0

Thus, the correct choice is:

D. r2cos⁡2θ−4rsin⁡θ=0r^2 \cos^2 \theta - 4r \sin \theta = 0

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Math Problem Analysis

Mathematical Concepts

Polar Coordinates
Rectangular Coordinates
Coordinate Geometry

Formulas

x = r cos(θ)
y = r sin(θ)
(r cos(θ))^2 = 4(r sin(θ))

Theorems

Coordinate Transformation Theorem

Suitable Grade Level

Grades 10-12

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