   Chapter 10.1, Problem 49E

Chapter
Section
Textbook Problem

Graph several members of the family of curves with parametric equations x = t + a cos t, y = t + a sin t, where a > 0. How does the shape change as a increases? For what values of a does the curve have a loop?

To determine

To plot: The graph members of family using the parametric equations x=t+acost and y=t+asint.

Explanation

Given data:

The parametric equation for the variable x is as follows.

x=t+acost (1)

The parametric equation for the variable y is as follows.

y=t+asint (2)

Calculation:

The parametric equation for the variable x is as follows.

x=t+acost

Substitute a=1 in equation (1),

x=t+(1)cost=t+cost

Substitute a=2 in equation (1),

x=t+(2)cost=t+2cost

Substitute a=3 in equation (1),

x=t+(3)cost=t+3cost

Substitute a=4 in equation (1),

x=t+(4)cost=t+4cost

The parametric equation for the variable y is as follows.

y=t+asint

Substitute a=1 in equation (2),

y=t+sint

Substitute a=2 in equation (2),

y=t+2sint

Substitute a=3 in equation (2),

y=t+3sint

Substitute a=4 in equation (2),

y=t+4sint

The value of t is increased from 1 to 4 with a step value of 1 and substituted in the parametric Equations x=t+cost and y=t+sint to obtain the value of x and y respectively.

Substitute 1 for t in Equation x=t+cost.

x=t+cost=(1)+cos(1)x=1.54

Substitute 1 for t in Equation y=t2+3t4.

y=t+sint=(1)+sin(1)y=1.84

The values of x and y for each step value of t is tabulated in the below table.

 t 1 2 3 4 x 1.54 0.58 0.01 0.34 y 1.84 2.9 3.14 3.24

The value of t is increased from 1 to 4 with a step value of 1 and substituted in the parametric equations x=t+2cost and y=t+2sint to obtain the value of x and y respectively.

Substitute 1 for t in Equation x=t+2cost.

x=t+2cost=(1)+2cos(1)x=2.08

Substitute 1 for t in Equation y=t+2sint

y=t+2sint=(1)+2sin(1)y=2.68

The values of x and y for each step value of t is tabulated in the below table.

 t 1 2 3 4 x 2.08 1.16 1.02 2.69 y 2.68 3.81 3.28 2.48

The value of t is increased from 1 to 4 with a step value of 1 and substituted in the parametric equations x=t+3cost and y=t+3sint to obtain the value of x and y respectively.

Substitute 1 for t in Equation x=t+3cost,

x=t+3cost=(1)+3cos(1)x=2

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