Question

Graph several members of the family of curves with parametric equations x=t+acos(t),y=t+asin(t), where a>0. How does the shape change as a increases? As a , the curves start having loops. The size of these loops increase with values of a. For what values of a does the curve have a loop? (Round your values to one decimal place. Enter your answer using interval notation.) ubmission 2 (0.66/1 points) Graph several members of the family of curves with parametric equations x=t+acos(t),y=t+asin(t), where a>0. How does the shape change as a increases? As a , the curves start having loops. The size of these loops increase with values of a. For what values of a does the curve have a loop?

Answer 1

As 'a' increases, the curves start having loops. The size of these loops increases with higher values of 'a'. For values of 'a' between 1 and the square root of 2 (inclusive), the curve has a loop.

Step-by-step explanation:

The given parametric equations are:

x = t + acos(t)
y = t + asin(t)

To graph the family of curves, we can fix a value for 'a' and plug in different values of 't' to obtain corresponding (x, y) pairs. By varying 'a', we can observe how the shape of the curves changes.

As 'a' increases, the curves start to exhibit loops. The size of these loops increases with higher values of 'a'.

To determine the values of 'a' for which the curve has a loop, we need to find the critical points where the slope of the curve is zero. These occur when dy/dt = 0.

Let's differentiate y with respect to t:
dy/dt = 1 + acos(t) + asin(t)

Setting dy/dt = 0:
1 + acos(t) + asin(t) = 0

Simplifying and solving for t:
acos(t) + asin(t) = -1
sqrt(2) * sin(t + π/4) = -1

Since sin(t + π/4) ranges from -1 to 1, the above equation has a solution only if -sqrt(2) <= -1 <= sqrt(2). Therefore, the curves have loops when 1 <= a <= sqrt(2).