Problem 7 - Calculus Early Transcendentals

Question

Sketch the following polar rectangles. $$R=\\{(r, \theta): 0 \leq r \leq 5,0 \leq \theta \leq \pi / 2\\}$$

Step-by-Step Solution

Answer

Question: Sketch the Polar Rectangle determined by the polar coordinates ranges 0 ≤ r ≤ 5 and 0 ≤ θ ≤ π/2. Answer: The polar rectangle is positioned in the first quadrant of the coordinate plane, with the lower-left vertex at the origin (0, 0) and dimensions 5 by 5. The rectangle is made up of sides AB, AD, BC, and DC with vertices A (0, 0), B (5, 0), C (0, 0), and D (0, 5).

Step 1: Identify the minimum and maximum range values for r and θ

In this problem, we are given the following ranges for the polar coordinates: - The distance r ranges from 0 to 5, i.e., 0 ≤ r ≤ 5 - The angle θ ranges from 0 to π/2, i.e., 0 ≤ θ ≤ π/2 Now, we want to sketch the polar rectangle for this given polar coordinate range.

Step 2: Convert the polar coordinate ranges to Cartesian coordinate ranges

To convert from polar coordinates to Cartesian coordinates, we can use the following formulas: - x = rcos(θ) - y = rsin(θ) In our case, the minimum value of r is 0 and the maximum value of r is 5. The minimum angle θ is 0 and the maximum angle θ is π/2. Therefore, let's find the Cartesian coordinate ranges. - For r = 0 and θ = 0: x(0,0) = 0 * cos(0) = 0, y(0,0) = 0 * sin(0) = 0, So we have point A: (x=0, y=0) - For r = 5 and θ = 0: x(5,0) = 5 * cos(0) = 5, y(5,0) = 5 * sin(0) = 0, So we have point B: (x=5, y=0) - For r = 0 and θ = π/2: x(0,π/2) = 0 * cos(π/2) = 0, y(0,π/2) = 0 * sin(π/2) = 0, So we have point C: (x=0, y=0) - For r = 5 and θ = π/2: x(5,π/2) = 5 * cos(π/2) = 0, y(5,π/2) = 5 * sin(π/2) = 5, So we have point D: (x=0, y=5)

Step 3: Sketch the rectangle based on the Cartesian coordinate ranges

Now, we have the Cartesian coordinates for the four corners of the rectangle: - Point A: (0, 0) - Point B: (5, 0) - Point C: (0, 0) - Point D: (0, 5) These points represent the vertices of the Cartesian rectangle. We can sketch the rectangle by drawing lines to connect these points. The result is a rectangle with dimensions 5 by 5, positioned in the first quadrant of the coordinate plane, with the lower-left vertex at the origin (0, 0). The rectangle has the following sides: AB, AD, BC, and DC.