Math Problem Statement

Test:MAC1114-T4 Question 1 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 Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation. r equals 4 Question content area bottom Part 1 What is the standard form of the equation in rectangular​ form?    enter your response here Part 2 What is the graph of this​ equation? A. horizontal line B. circle with center​ (0, 0), radius 16 C. circle with center​ (0, 0), radius 4 D. vertical line Part 3 Select the graph of r equals 4. A. -8 -4 4 8 -8 -4 4 8

A polar grid has concentric circles around the pole, with radii r=2 to r=10 in increments of 2, intersecting 24 evenly spaced unlabeled rays from the pole in increments of StartFraction pi Over 12 EndFraction. A circle is plotted on the second concentric circle from the pole with radius 4. All coordinates are given in polar coordinates. B. -8 -4 4 8 -8 -4 4 8

A polar grid has concentric circles around the pole, with radii r=2 to r=10 in increments of 2, intersecting 24 evenly spaced unlabeled rays from the pole in increments of StartFraction pi Over 12 EndFraction. A circle is plotted on the first concentric circle from the pole with radius 2. All coordinates are given in polar coordinates. C. -8 -4 4 8 -8 -4 4 8

A polar grid has concentric circles around the pole, with radii r=2 to r=10 in increments of 2, intersecting 24 evenly spaced unlabeled rays from the pole in increments of StartFraction pi Over 12 EndFraction. A horizontal line passes through the point (4, StartFraction pi Over 2 EndFraction). All coordinates are given in polar coordinates. D. -8 -4 4 8 -8 -4 4 8

A polar grid has concentric circles around the pole, with radii r=2 to r=10 in increments of 2, intersecting 24 evenly spaced unlabeled rays from the pole in increments of StartFraction pi Over 12 EndFraction. A vertical line passes through the point (4, 0). All coordinates are given in polar coordinates. What is the standard form of the equation in rectangular form?What is the graph of this equation?Select the graph of r equals 4. Time Remaining: 02:29:03

Solution

To solve this problem, let’s carefully analyze each part.

Part 1: Transform the polar equation r=4r = 4 to rectangular form

In polar coordinates, the relationship between rr and rectangular coordinates xx and yy is given by: r=x2+y2.r = \sqrt{x^2 + y^2}.

For r=4r = 4: x2+y2=4.\sqrt{x^2 + y^2} = 4.

Squaring both sides: x2+y2=16.x^2 + y^2 = 16.

This is the standard equation of a circle in rectangular coordinates, with center (0,0)(0, 0) and radius 44.

Part 2: Identify the graph

The equation x2+y2=16x^2 + y^2 = 16 represents a circle with:

  • Center: (0,0)(0, 0)
  • Radius: 44

So, the correct option is: C. Circle with center (0, 0), radius 4.

Part 3: Select the graph of r=4r = 4

From the descriptions:

  • A circle with radius 4 is plotted on the second concentric circle from the pole.
  • Correct answer: A (the graph matches a circle of radius 4 centered at the pole).

Final Answers:

  1. Standard form in rectangular coordinates: x2+y2=16x^2 + y^2 = 16
  2. Graph of this equation: Circle with center (0,0)(0, 0), radius 44 (C).
  3. Select the graph: A.

Would you like a detailed breakdown of any part?

Related Questions:

  1. How do you derive the formula x2+y2=r2x^2 + y^2 = r^2 from polar coordinates?
  2. What transformations apply to polar equations when shifting the center?
  3. How does the graph of r=cr = c change for different constants cc?
  4. How can you verify the radius of a circle given its rectangular equation?
  5. What happens to the equation if rr is negative?

Tip:

Always square both sides carefully when converting from polar to rectangular forms, ensuring you interpret r2=x2+y2r^2 = x^2 + y^2 correctly.

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Math Problem Analysis

Mathematical Concepts

Polar Coordinates
Rectangular Coordinates
Circle Equations

Formulas

r = √(x² + y²)
x² + y² = r²

Theorems

Equation of a circle in rectangular coordinates

Suitable Grade Level

Grades 9-12

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