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Chapter 12: Problem 26

Find an equation or inequality that describes the following objects. A ball with center (0,-2,6) with the point (1,4,8) on its boundary

Short Answer

Expert verified
Answer: The equation or inequality describing the ball is x^2 + (y + 2)^2 + (z - 6)^2 鈮� 41.

Step by step solution

01

Compute the radius of the ball

To calculate the radius of the ball, we'll find the distance between the center and the point on the boundary: r = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2) Here, (x_1, y_1, z_1) are the coordinates of the center and (x_2, y_2, z_2) are the coordinates of the boundary point. r = sqrt((1 - 0)^2 + (4 - (-2))^2 + (8 - 6)^2)
02

Calculate the radius

Now, plug in the coordinates of the points and calculate the radius: r = sqrt((1 - 0)^2 + (4 - (-2))^2 + (8 - 6)^2) r = sqrt(1^2 + 6^2 + 2^2) r = sqrt(1 + 36 + 4) r = sqrt(41) The radius of the ball is sqrt(41).
03

Write the equation of the ball

The equation of a sphere (and by extension, the ball) in 3D space is given by: (x - x_c)^2 + (y - y_c)^2 + (z - z_c)^2 鈮� r^2 where (x_c, y_c, z_c) are the coordinates of the center and r is the radius of the ball. In this case, the center is (0, -2, 6) and the radius is sqrt(41). Plugging in these values, we get the equation of the ball: (x - 0)^2 + (y - (-2))^2 + (z - 6)^2 鈮� (sqrt(41))^2 x^2 + (y + 2)^2 + (z - 6)^2 鈮� 41 This is the equation or inequality that describes the ball in the 3-dimensional coordinate system.

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