Chapter 1. calc_tutorial_13_1_014

1.1 Problem Statement

{2,3,4,5}
{11,13,15,17}
pow($b,2)
{0}

The function r(t) traces a circle. Determine the radius, center of the circle, and plane containing the circle.

r(t) = $ai + ($b cos t)j + ($b sin t)k

1.2 Step 1

Question Sequence

Question 1.1

In order to determine the radius, center, and plane containing the circle, we must write the components of r(t) as parametric functions in (x, y, z). Given r(t) = $ai + ($b cos t)j + ($b sin t)k, write the appropriate parametric equations describing x(t), y(t), z(t).

x(t) = nc1ItEz0kR4=

y(t) = iSba6t70dtA=cos t

z(t) = iSba6t70dtA=sin t

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

To determine the radius of the circle, calculate y2(t) + z2(t) since y(t) and z(t) both contain the trigonometric functions which will parametrize the circle.

y2(t) + z2(t) = ($b cos t)2 + ($b cos t)2 = $c

State the radius of the circle parametrized by r(t).

radius = iSba6t70dtA=

Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Since x(t) = $a is a constant function, every point on the circle has x-component $a. This fixes the plane containing the circle.

Determine the plane containing the circle.

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Correct.
Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

Since the circle lies in the plane x = $a, the x-component of the center of the circle must be

x = nc1ItEz0kR4=

To determine the y- and z-components of the center of the circle defined by r(t), look at y2 + z2 = $b2 from step 2. This is the equation of a circle with radius $b, centered at

(y, z) = (U2GIbglD1oM=, U2GIbglD1oM=)

Adding the third component, x = $a, does not affect the y- and z-components corresponding to the center of this circle.

State the center of the circle defined by r(t).

(x, y, z) = (nc1ItEz0kR4=, U2GIbglD1oM=, U2GIbglD1oM=)

Correct.
Incorrect.